Inclusion Criteria:  Aged 18 to 80 years at time of enrollment Able to provide a recent tumor specimen taken since the time of the subject's most recent systemic anticancer therapy, for GPC3 expression assessment by IHC. Histologically confirmed advanced unresectable or metastatic hepatocellular carcinoma (HCC), squamous cell carcinoma (SCC) of the lung, liposarcoma, or Merkel cell carcinoma (MCC) with GPC3 overexpression by IHC. Subjects must consent to IHC testing in a separate informed consent. Note: Tumor samples will be sent to a central laboratory for GPC3 expression analysis. Documentation of disease progression or refractory disease or intolerance to prior lines of standard-of-care (SOC) therapies. Patients with tumors with genetic alterations and mutations (eg, breast cancer gene [BRCA], epidermal growth factor receptor [EGFR] mutations, and anaplastic lymphoma kinase [ALK] translocation) who have approved targeted therapies available for their cancer will need to have been treated with such approved therapies or refused such approved targeted therapy for their cancer prior to enrolling in this study. Life expectancy >16 weeks Have adequate organ/renal function Left ventricular ejection fraction (LVEF) ≥50% by multiple-gated acquisition (MUGA) scan or echocardiogram (ECHO) Eastern Cooperative Group performance status of 0 to 1  For subjects with HCC:  Child-Pugh Score of A No fibrolamellar carcinoma or mixed hepatocellular cholangiocarcinoma histology No moderate or severe ascites A minimum of 2 sites of disease, including at least 1 site that is measurable by RECIST Version 1.1 criteria to ensure sufficient disease for response assessment. At least 1 of the other lesions must be considered adequate for protocol-required tumor biopsy. Consider prioritization of percutaneous lesions that are palpable, or guided by imaging if necessary, and exclude biopsies of any lesions that are in proximity to vital visceral, cardio-pulmonary, or any neurovascular structures. A single site of disease is considered adequate to allow for response assessment and protocol-required tumor biopsies if it measures at least 2 cm in the shortest axis).  Adequate wash-out of prior systemic therapy for underlying malignancy, relative to leukapheresis:  Last dose of any antineoplastic treatment must be at least 2 weeks before leukapheresis. Last dose of any investigational agent must be at least 3 half-lives of the treatment, or 28 days, before leukapheresis (whichever is shorter). Adequate wash-out of prior systemic therapy for underlying malignancy, relative to LD chemotherapy: The last dose of systemic nitrosourea or systemic mitomycin-C must be at least 6 weeks before the first dose of LD chemotherapy in this study. For all other for systemic cytotoxic chemotherapy, the wash-out must be the duration of a full cycle of that therapy. For example, if the prior therapy is given in 3-week cycles, there must be at least 3 weeks between the last dose of that therapy and the first dose of LD chemotherapy in this study. If the prior therapy is administered more frequently than every 2 weeks, a minimum 2-week wash-out is required. For biologic therapy (eg, antibodies) the wash-out must be either the duration of the biologic agent dosing interval, or 3 weeks, whichever is shorter. For small molecule therapies, the wash-out must be 5 half-lives of the drug. For any investigational agent, the wash-out must be 3 half-lives of the investigational agent, or 4 weeks, whichever is shorter.  Note: Local radiation of lesions is allowed if indicated for palliation; locally treated lesions will be considered non-target lesions. Hormone ablation is also allowed as clinically indicated.  Subjects or their legally acceptable representative must be able and willing to:  Provide IRB/IEC-approved written informed consent in accordance with regulatory and institutional guidelines. This must be obtained before the performance of any protocol related procedures that are not part of normal patient care. Comply with the study protocol and with the planned biopsy procedures. Willing and able to commit to study assessments and visit schedule, including availability of a caregiver to conduct daily neurological assessments for 28 days after BOXR1030 administration. For women of childbearing potential (defined as physiologically capable of becoming pregnant), confirmation of a negative serum pregnancy test and agreement to use of highly effective contraception per Clinical Trials Facilitation and Coordination Group (CTFG) criteria from screening through the first 12 months after BOXR1030 administration. For men with partners of childbearing potential, agreement to use effective barrier contraception from screening through the first 12 months after BOXR1030 administration.  Exclusion Criteria:  Prior treatment with adoptive cell therapy (eg, CAR T-cell therapy, natural killer cell therapy, engineered T-cell receptor therapy). History of allogenic hematopoietic stem cell transplant (HSCT). Untreated central nervous system (CNS) tumors or brain metastasis. Patients are eligible if CNS metastases are asymptomatic and have been treated and patients have neurologically returned to baseline (residual signs or symptoms related to the CNS treatment are permitted). Imaging obtained for the purpose of CNS metastases management performed within 28 days prior to Day 1 must document radiographic stability of CNS lesions and be performed after completion of any CNS directed therapy. CNS evaluation for asymptomatic patients is not required for the study. Patients with leptomeningeal metastases are excluded. Patients who have not recovered to ≤ Grade 1 or baseline from all AEs due to previous therapies (patients with ≤ Grade 2 neuropathy that has been stable for at least 4 weeks or ≤ Grade 2 endocrine-related AEs that has been stable for at least 4 weeks on replacement therapy). Planned use of any antineoplastic treatment or investigational agent from the time of the first dose of LD chemotherapy through the end of study participation, except for allowed local radiation of lesions for palliation (to be considered non-target lesions after treatment) and hormone ablation.  Uncontrolled or life-threatening symptomatic concomitant disease including clinically significant gastrointestinal bleeding or pulmonary hemorrhage within 4 weeks before screening, known symptomatic human immunodeficiency virus (HIV) positive with an acquired immunodeficiency syndrome (AIDS)-defining opportunistic infection within the last year, or a current CD4 count <350 cells/uL, symptomatic active hepatitis B or C checked at screening, or active tuberculosis. Patients with HIV are eligible if:  They have received antiretroviral therapy (ART) as clinically indicated for at least 4 weeks prior to starting study treatment They continue on ART as clinically indicated while enrolled on study CD4 counts and viral load are monitored per standard of care by a local health care provider Has undergone a major surgery within 3 weeks of starting study treatment or has inadequate healing or recovery from complications of surgery prior to starting study treatment. Has received prior radiotherapy within 2 weeks of start of study treatment. Subjects must have recovered from all radiation-related toxicities, not require corticosteroids, and not have had severe radiation pneumonitis. A 1-week wash-out is permitted for palliative radiation (≤2 weeks of radiotherapy) to non-CNS disease. Potentially life-threatening second malignancy requiring systemic treatment within the last 3 years (ie, patients with a history of prior malignancy are eligible if treatment was completed at least 3 years before entering the Treatment Period and the patient has no evidence of disease) or which would impede evaluation of treatment response. Hormone ablation therapy is allowed within the last 3 years. Patients with history of prior early stage basal/squamous cell skin cancer or non-invasive or in situ cancers that have undergone definitive treatment at any time are eligible. Clinically significant (ie, active) cardiovascular disease: cerebral vascular accident/stroke (<6 months prior to enrollment), myocardial infarction (<6 months prior to enrollment), unstable angina, congestive heart failure (≥ New York Heart Association Classification Class II), or the presence of any condition that can increase proarrhythmic risk (eg, hypokalemia, bradycardia, heart block) including any new, unstable, or serious cardiac arrhythmia requiring medication, or other baseline arrhythmia that might interfere with interpretation of electrocardiograms (ECGs) on study (eg, bundle branch block). Patients with QTcF >450 msec for males and >470 msec for females on screening ECG are excluded. Any patients with a bundle branch block will be excluded with QTcF >450 msec. Males who are on stable doses of concomitant medication with known prolongation of QTcF (eg, selective serotonin re-uptake inhibitor antidepressants) will only be excluded for QTcF >470 msec. Has an active infection requiring systemic therapy. A woman of child-bearing potential (WOCBP) who has a positive pregnancy test prior to treatment. Is breastfeeding or expecting to conceive or father children within the projected duration of the study, starting with the Screening Visit through 6 months after the last dose of study treatment. Unable to receive any of the agents used in this study due to history of severe immediate hypersensitivity reaction (eg, hypersensitivity to dimethyl sulfoxide [DMSO]). Known allergy or contraindication to any of the LD chemotherapy or prophylaxis medications required during the study.  Any other significant co-morbid disease or condition which, in the judgment of the Investigator, would put the subject at undue risk (eg, cirrhotic liver disease).  -