Inclusion Criteria:  PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Patients must have histologically or cytologically confirmed NC based on at least one of the following criteria:  Ectopic expression of NUT protein (> 50% tumor nuclei) as determined by immunohistochemistry (IHC) testing performed in a Clinical Laboratory Improvement Act (CLIA) certified laboratory Detection of the NUT gene translocation as determined by fluorescence in situ hybridization (FISH) performed at the Bringham and Women's Hospital (BWH) Center for Advanced Molecular Diagnostics (CAMD) PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Participants must have disease that is metastatic, unresectable, or for which a surgical approach would not likely confer a survival benefit or would be otherwise contraindicated. Participants who have already undergone surgical resection are eligible. PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Age >= 12 years. Patients 12-17 years of age must be >= 40 kg at enrollment. Because no dosing or adverse event data are currently available on the use of ZEN003694 in combination with etoposide and cisplatin in patients < 12 years of age, younger children are excluded from this study. PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Eastern Cooperative Oncology Group performance status of =< 2 (Karnofsky >= 60%) for patients >= 16 years of age, Lansky >= 50% if < 16 years of age PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Participants must have measurable disease per Response Evaluation Criteria in Solid Tumors (RECIST) version 1.1 criteria. Patients in the phase 1 portion do not need measurable disease if their disease is otherwise evaluable. PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Ability to swallow and retain oral medication. PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Absolute neutrophil count >= 1.5 x 10^9/L PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Platelets >= 125 x 10^9/L PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Hemoglobin >= 9.0 g/dL PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Albumin >= 2.5 g/dL PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Total bilirubin =< 1.5 x institutional upper limit of normal (ULN) for age PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Aspartate aminotransferase (AST)(serum glutamic-oxaloacetic transaminase [SGOT])/alanine aminotransferase(ALT)(serum glutamate pyruvate transaminase [SGPT]) =< 2.5 x institutional ULN for age PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Calculated creatinine clearance >= 50 mL/min (using the CKD-epi equation for adults or Schwartz formula for patients 12-17 years) PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Prothrombin time/international normalized ratio =< 1.5 x ULN PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Partial thromboplastin time =< 1.5 x ULN PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: QT interval by Fridericia (QTcF) < 450 ms (machine or manual read allowed). Patients should avoid medications which prolong the QT. PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 3 months are eligible for this trial as long as their anti-retroviral therapy does not have the potential for drug-drug interactions as judged by the treating investigator. PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated. PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load. Hepatitis C (HepC antibody) testing is required. Hepatitis C ribonucleic acid (RNA) is optional; however, a confirmatory negative Hepatitis C RNA test must be obtained to be able to enroll participants with positive Hepatitis C antibody due to prior resolved disease. PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for treatment in the phase 1 portion, but not in the phase 2 or non-thoracic, non-BRD4 exploratory cohort. Participants enrolling to the phase 2 or non-thoracic, non-BRD4 exploratory cohort with a history of prior malignancy are eligible only if they fit one or more of the following criteria: participants with non-melanoma skin cancers that have been curatively treated; participants with adequately treated carcinomas in situ of any type; or participants who were diagnosed and curatively treated at least 3 years prior to the date of study entry and who are considered by the treating investigator to be at low risk for recurrence.  PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: The effects of ZEN003694 on the developing human fetus are unknown. For this reason and because the chemotherapeutic agents used in this trial are known to be teratogenic, women of child-bearing potential and men must agree to use adequate contraception (barrier method of birth control; abstinence) prior to study entry and for the duration of study participation and for 4 months after completion of study therapy. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. Men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 4 months after completion of study therapy. For female subjects of child-bearing potentially receiving ZEN003694, hormonal means of birth control alone, such as oral, injectable, dermal, subdermal or topical contraceptives are NOT acceptable forms of birth control given that their efficacy has not been evaluated when given in combination with the investigational drugs. "Adequate contraception" is defined as the following: Contraceptive methods with a failure rate of =< 1% used in combination with the barrier method. The following contraceptive methods are acceptable to use in combination with the barrier method: intrauterine device (IUD), intrauterine system (IUS), or oral contraceptive pills (OCPs) that meet the < 1% failure rate as stated in the product label. Note: Hormonal IUDs/OCPs may only be used if the following criteria are met: male condoms are required AND subjects are informed of the potential for reduced systemic hormone levels from the IUD/OCP when taking ZEN003694. Alternatively, male partner sterilization (vasectomy with documentation of azoospermia) prior to the female subject's entry into the study, and this male is the sole partner for that subject. For this definition, "documented" refers to the outcome of the investigator's/designee's medical examination of the subject or review of the subject's medical history for study eligibility, as obtained via a verbal interview with the subject or from the subject's medical records. Male subjects with female partners of child-bearing potential must use one of the following contraceptive methods:  Vasectomy with documentation of azoospermia OR Condom use PLUS partner use of a highly effective contraceptive (=< 1% rate of failure per year) such as intrauterine device or system, or hormonal birth control such as contraceptive subdermal implant, combined estrogen and progestogen oral contraceptive, injectable progestogen, contraceptive vaginal ring, or percutaneous contraceptive patches. PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Male subjects should not donate sperm while on study and for 16 weeks after the last dose of study medication. Male subjects whose partners are or become pregnant must continue to use condoms for 16 weeks after the last dose of study medication. PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Ability to understand and the willingness to sign a written informed consent document (or parent or legally authorized representative if minor). Participants with impaired decision-making capacity (IDMC) who have a legally-authorized representative (LAR) and/or family member available may be eligible after discussion with the Principal Investigator of this study. PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Women of childbearing potential must have a negative pregnancy test within 7 days of starting treatment. PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Participants who have not had cytotoxic chemotherapy, oral tyrosine kinase inhibitor (TKI) or small molecule therapy, or immunotherapy within 2 weeks prior to the first dose of study medication or 5 half-lives, whichever is shorter. There is no required washout for previous EP therapy. Patients may have had a maximum of 2 prior cycles of EP. There is no required washout for previous ZEN003694 therapy for patients enrolling to the dose escalation or phase 2 portion of the study. PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Participants who have received prior radiation therapy can be enrolled at least 1 week after completing radiation. PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Participants who have had major surgery can be enrolled at least 3 weeks after the surgery. PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Any therapy-related toxicities must have resolved to =< grade 1 or baseline (with the exception of alopecia, peripheral neuropathy, or rash that will be permitted at =< grade 2). Other grade 2 toxicities attributed to prior treatment may be permitted with agreement from the overall principal investigator if they are toxicities not commonly attributed to ZEN003694. Toxicities attributed to current EP therapy are excluded from this requirement for participants enrolling to the dose escalation or phase 2 portion of the study, as long as the participant meets all other eligibility criteria. NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Participants must lack BRD4-NUT rearrangement as identified via FISH testing performed at the BWH CAMD or by confirmation of a specific non-BRD4-NUT rearrangement by a next-generation sequencing (NGS) assay. NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Patients who have had a bone-targeted radionuclide more than 6 weeks before the first dose of ZEN003694 are eligible. NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Patients who have had major surgery other than diagnostic surgery, dental surgery or stenting more than 3 weeks prior to the first dose of ZEN003694.  Exclusion Criteria:  PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Participants with known untreated central nervous system (CNS) metastases. Patients with a history of treated CNS metastases are eligible, provided they meet the following criteria:  Radiologically stable for 4 weeks. Disease outside the CNS is present. Recovery from acute toxicity associated with the treatment to =< Common Terminology Criteria for Adverse Events (CTCAE) grade 1 or baseline (with the exception of alopecia), with no requirement for escalating doses of corticosteroids over the past 7 days. Subjects currently taking enzyme-inducing anticonvulsants (EIAC) must be transitioned to non-enzyme inducing anticonvulsants at least 14 days or 5 half-lives prior to the first dose of study medication No presence of symptomatic or untreated leptomeningeal metastases or spinal cord compression PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: History of allergic reactions attributed to compounds of similar chemical or biologic composition to ZEN003694 or other agents used in study. PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Uncontrolled intercurrent illness including, but not limited to: ongoing or active infection requiring systemic treatment, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements. Patients with myocardial infarction or unstable angina within 6 months prior to the first dose of ZEN003694 will be excluded. PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Any gastrointestinal (GI) disorder that may affect absorption of ZEN003694 and other oral medications in the opinion of the treating investigator, such as malabsorption syndrome or major bowel or stomach resection. PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Patients who are receiving any other investigational agents. PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Patients receiving any medications or substances that are strong inhibitors or inducers of CYP3A4 are ineligible. Strong inhibitors or inducers of CYP3A4 must be discontinued at least 7 days prior to the first dose of ZEN003694. Because the lists of these agents are constantly changing, it is important to regularly consult a frequently-updated medical reference. As part of the enrollment/informed consent procedures, the patient will be counseled on the risk of interactions with other agents, and what to do if new medications need to be prescribed or if the patient is considering a new over-the-counter medicine or herbal product. PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Pregnant women are excluded from this study because ZEN003694 is a BET inhibitor with the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with ZEN003694, breastfeeding should be discontinued if the mother is treated with ZEN003694. These potential risks may also apply to other agents used in this study. PHASE 1, PHASE 2, AND NON-THORACIC, NON-BRD4 EXPLORATORY COHORT: Patients receiving therapeutic-dose anticoagulation (e.g., warfarin, low-molecular weight heparin [LMWH], or novel oral anticoagulants) are not eligible. Use of oral Factor Xa inhibitors (i.e., rivaroxaban, apixaban, betrixaban, edoxaban otamixaban, letaxaban, eribaxaban) and Factor IIa inhibitors (i.e., dabigatran) is not permitted. Prophylactic anticoagulation, with low doses (per stan