This study has two phases. The objective of Phase I to find the maximum dose (MTD) of Ruxolitinib when combined with standard dose of paclitaxel given weekly for advanced or metastatic breast cancer. Three participants will be entered at a dose ruxolitinib equaling 10 mg orally twice daily with weekly paclitaxel. If no dose-limiting toxicity is seen after 6 weeks of treatment (two cycles), then the dose of ruxolitinib will be escalated using a standard 3+3 design, until 2 participants experience dose limiting toxicity (DLT). The dose below the DLT is designated the MTD and this dose of ruxolitinib will be used in the phase II preoperative study for triple negative IBC.

During Cycle 1 the participant will come into clinic every week. At each visit, the participant will have a physical exam and will be asked questions regarding general health and specific questions about any problems that the participant might be having with any medications. About 2-3 additional tablespoons of blood will be taken before the participant's begins ruxolitinib, on Cycle 2 Day 1, Cycle 3 Day 1, and at the end of the study for research blood tests. Because these tests are being performed for research, and their clinical usefulness is unknown, the participant will not receive the results of these tests. The investigator will assess the participant's tumor by CT scans or MRI every 2 cycles. In addition, if the participant has tumors that are visible or can be palpated (felt), then they will be measured by the participant's study doctor in the clinic. If the participant has had a history of cancer in the bones or suspected cancer in the bones, then a bone scan will be performed before the participant can begin ruxolitinib. The bone scan may be repeated every 2 cycles and at the end of the study if the participant study doctor believes it is clinically needed. Otherwise, it does not need to be repeated. Photographs may be taken of the participant's tumor to assess the tumor response to the treatment.

The phase II period of the study will treat triple negative inflammatory breast cancer participants with ruxolitinib combined with 12 weeks of weekly paclitaxel followed by standard care Doxorubicin and Cyclophosphamide (AC) chemotherapy, eligible participants will proceed to surgical mastectomy followed by radiation. The phase II study will begin once the phase I study has been completed.

During the phase II study, participants will have a research biopsy of the breast followed by one week of ruxolitinib given twice daily. A second research biopsy of the breast is then performed and the participants will then receive combination ruxolitinib (at the MTD dose defined in the phase I study) and standard dose paclitaxel for 12 weeks. Participants will be seen weekly during treatment. One week after completing the combination therapy, participants will receive standard dose doxorubicin and cyclophosphamide (AC) every 2 weeks for 4 cycles.

We will evaluate the effect of JAK inhibition by ruxolitinib on the tumor by comparing pSTAT3+ expression of the pre-treatment research biopsy with the pSTAT3+ expression on the second research biopsy performed after one week of ruxolitinib. Participants who have disease regression following 12 weeks of ruxolitinib and paclitaxel followed by AC chemotherapy will undergo mastectomy, and the amount of residual breast cancer will be assessed. We will correlate the degree of residual cancer in the mastectomy with the amount of pSTAT3+ expression seen in the research biopsies. We will be checking standard blood tests, IL-6 and CRP levels throughout the treatment to see if the levels change in response to ruxolitinib treatment. This may be an easier method of determining treatment efficacy. Standard radiation therapy will be given following mastectomy.