Study of Itacitinib for the Prophylaxis of Graft-Versus-Host Disease and Cytokine Release Syndrome After T-cell Replete Haploidentical Peripheral Blood Hematopoietic Cell Transplantation

Inclusion Criteria: Patients must meet the following criteria within 30 days prior to Day 0 unless otherwise noted. * Diagnosis of a hematological malignancy listed below: * Acute myelogenous leukemia (AML) in complete morphological remission (based on International Working Group (IWG) Criteria) * Acute lymphocytic leukemia (ALL) in complete morphological remission (MRD negative, based on IWG Criteria) * Myelodysplastic syndrome with ≤ 5% blasts in bone marrow. * Non-Hodgkin's lymphoma (NHL) or Hodgkin's disease (HD) in 2nd or greater complete or partial remission. * Planned treatment is myeloablative or reduced intensity conditioning followed by T Cell-replete peripheral blood haploidentical donor transplantation * Available human leukocyte antigen (HLA)-haploidentical donor who meets the following criteria: * Blood-related family member, including (but not limited to) sibling, offspring, cousin, nephew, or parent. Younger donors should be prioritized. * At least 18 years of age * HLA-haploidentical donor/recipient match by at least low-resolution typing per institutional standards. * In the investigator's opinion, is in general good health, and medically able to tolerate leukapheresis required for harvesting hematopoietic stem cells (HSC). * No active hepatitis. * Negative for human T-cell lymphotrophic virus (HTLV) and human immunodeficiency virus (HIV). * Not pregnant. * Safety Lead-In Phase: For the first three patients, the donor must consent to a second product collection should it prove necessary. * Eastern Cooperative Oncology Group (ECOG) performance status ≤ 2 * Adequate organ function as defined below: * Total bilirubin must be within normal range at baseline * Aspartate aminotransferase (AST)(SGOT) and alanine aminotransferase (ALT) (SGPT) ≤ 3.0 x institutional upper limit of normal (IULN). * Creatinine ≤ 1.5 x IULN OR creatinine clearance ≥ 45 mL/min/1.73 m\^2 by Cockcroft-Gault Formula. * Oxygen saturation ≥ 90% on room air. * Left ventricular ejection fraction (LVEF) ≥ 40%. * Forced expiratory volume (FEV1) and forced vital capacity (FVC) ≥ 40% predicted, diffusing capacity of the lung for carbon monoxide (DLCOc) ≥ 40% predicted. If DLCO is \< 40%, patients will still be considered eligible if deemed safe after a pulmonary evaluation. * At least 18 years of age at the time of study registration * Able to understand and willing to sign an Institutional Review Board (IRB) approved written informed consent document (or that of legally authorized representative, if applicable). * Must be able to receive GVHD prophylaxis with tacrolimus, mycophenolate mofetil, and cyclophosphamide Exclusion Criteria: * Must not have undergone a prior allogeneic donor (related, unrelated, or cord) transplant. Prior autologous transplant is not exclusionary. * Presence of donor-specific antibodies (DSA) with Mean Fluorescence Intensity (MFI) of ≥2000 as assessed by the single antigen bead assay. * Known HIV or active hepatitis B or C infection. * Known hypersensitivity to one or more of the study agents, including Ruxolitinib and Itacitinib. * Must not have myelofibrosis (unless they are enrolled Amendment #5 or later) or other disease known to prolong neutrophil engraftment to \> 35 days after transplant. * Must not receive antithymocyte globulin as part of pre-transplant conditioning regimens. * Currently receiving or has received any investigational drugs within the 14 days prior to the first dose of study drug (Day -3). * Pregnant and/or breastfeeding. * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, autoimmune disease, symptomatic congestive heart failure, unstable angina pectoris, unstable cardiac arrhythmias, or psychiatric illness/social situations that would limit compliance with study requirements. * Immunosuppressive doses of steroids. Subjects with steroids for adrenal insufficiency will not be excluded. Additional Inclusion Criteria Under Amendment 5 * Five subjects with myelofibrosis will be enrolled in the expansion phase. * Three patients whose donors fail to collect the target number of CD34+ cells and the treating physician choses to move forward with the haplo-HCT will be enrolled in the expansion phase.