Brief Summary: The overall objective of this protocol is to improve the cure rate of relapsed precursor B-cell acute lymphoblastic leukemia (ALL) and lymphoblastic lymphoma. This phase II trial is studying risk-directed therapy for B-lymphoblastic leukemia or lymphoma in first relapse. Standard risk (SR) and high risk (HR) participants will receive different therapy. Treatment will consist of chemotherapy for SR participants, and chemotherapy followed by hematopoietic stem cell transplant (HSCT) for HR in first relapse. Induction therapy consists of three blocks of chemotherapy. The first block is a novel immunotherapy regimen that includes chemotherapy, rituximab and infusion of haploidentical natural killer (NK) cells. SR participants will continue to receive chemotherapy for a total duration of approximately 2 years. HR participants will be candidates for HSCT and will proceed to transplant once a suitable donor is found and their minimal residual disease (MRD) is negative.

Detailed Description: Primary objective * To estimate the 3-year survival rate of participants with first relapse or primary refractory precursor B-cell ALL and lymphoblastic lymphoma treated with risk-directed therapy. Secondary objectives * To determine minimal residual disease (MRD) levels at the end of remission induction therapy for participants with relapsed precursor B-cell ALL and compare the results with those in protocol ALLR17 * To estimate levels of CD20 expression at baseline, during treatment with dexamethasone-containing chemotherapy and following rituximab treatment in Block 1 of remission induction therapy for relapsed precursor B-cell ALL. STUDY DESCRIPTION: The general treatment plan will consist of chemotherapy for standard-risk participants and chemotherapy followed by HSCT for high risk participants in first relapse of B-precursor ALL or lymphoblastic lymphoma. Remission induction for all participants consists of three blocks of therapy, wherein the first block is a novel immunotherapy regimen that includes cytotoxic chemotherapy, rituximab and infusion of haploidentical natural killer (NK) cells. Standard-risk patients will continue to receive chemotherapy for a total duration of approximately 2 years. High-risk patients will be candidates for HSCT and will proceed to transplant once a suitable donor is found and the patient achieves negative MRD. Participants will be assigned to the standard arm if they experience late relapse (\> or = 6 months after completion of frontline therapy) AND maximum residual disease (MRD) is \<0.01% at the end of Block II of remission induction therapy. Provisional standard risk participants (i.e., late relapse) will be re-assigned to high risk if MRD is \> or = 0.01% at the end of Block II. Participants with lymphoma must be in complete remission at the end of Block III. High risk participants will meet one of the following criteria: * Early relapse (on therapy or \<6 months after completion of frontline therapy), OR * Any relapse after hematopoietic stem cell transplant, OR * MRD \> or = 0.01% at the end of Block II of remission induction therapy, OR * Re-emergence of MRD at any time after attaining negative MRD on this clinical trial. Natural killer (NK) cell collection: Donors who meet eligibility criteria will undergo apheresis once. The cells obtained will be purified for CD56+ cells utilizing the CliniMACS selection system. The NK cell product will undergo quality control testing following standard operating procedures of the St. Jude Human Applications Laboratory. OUTLINE (STANDARD RISK): REMISSION INDUCTION: BLOCK I: Patients receive dexamethasone orally (PO) or intravenously (IV) thrice daily (TID) on days 1-8 and 21-28; vincristine sulfate IV on days 1, 21, 28, and 35; rituximab IV on days 4, 13, 20, and 27; clofarabine, cyclophosphamide, and etoposide IV on days 6-10; aldesleukin subcutaneously (SC) once every other day (QOD) on days 11-19; and pegaspargase IV on days 21 and 35. Patients also undergo natural killer (NK) cell infusion on day 12. Patients may receive triple intrathecal therapy comprising methotrexate, therapeutic hydrocortisone, and cytarabine on days 1, 5, 8, 11, 21, and 28. Patients continue on to Block II after count recovery. BLOCK II: Patients receive methotrexate IV over 24 hours on days 1 and 8 and mercaptopurine PO on days 1-21. Patients also receive triple intrathecal therapy on day 1. High-risk patients with negative MRD continue on to transplantation. All patients with positive MRD continue on to Block III after count recovery. BLOCK III: Patients receive cytarabine IV over 2 hours twice daily (BID) on days 1-4 and mitoxantrone hydrochloride IV over 1 hour on days 3-5. Patients also receive triple intrathecal therapy on day 7. INTERIM CONTINUATION (for patients unable to tolerate dose intensive chemotherapy): Patients receive etoposide and cyclophosphamide IV on day 1, methotrexate IV on day 8, mercaptopurine PO on days 8-14, teniposide and cytarabine IV on day 15, dexamethasone PO TID on days 22-26, and vinblastine IV on day 22. RE-INDUCTION THERAPY: Patients receive clofarabine, cyclophosphamide, and etoposide IV on days 1-5; dexamethasone PO on days 1-6; and pegaspargase on days 6 and 20 and vincristine sulfate IV on days 6, 13, and 20. Patients may also receive triple intrathecal therapy on days 1 and 15. Patients continue on to continuation treatment after count recovery. CONTINUATION TREATMENT: Patients receive methotrexate IV over 2 hours on day 1 and mercaptopurine PO on days 1-7 of weeks 1, 2, 5, and 6; teniposide and cytarabine IV on day 1 of weeks 3 and 7; vincristine sulfate IV on day 1 of week 4; dexamethasone PO TID on days 1-5 of weeks 4 and 8; and vinblastine IV on day 1 of week 8. Treatment repeats every 8 weeks for up to 10 courses in the absence of disease progression or unacceptable toxicity. Patients may also receive triple intrathecal therapy on day 1 of week 1 of all courses and day 1 of week 5, courses 1-5. Due to the unavailability of the drug Teniposide (VM-26), etoposide (VP-16) will be substituted for the remaining doses for patients currently on this study. The protocol Section 5.1.3 allows this substitution. OUTLINE: GROUP 2 (high risk defined as participants not meeting the standard risk criteria noted above): Patients receive Remission Induction (Blocks I, II, and III) treatment as described above for Group 1. Patients then undergo allogeneic hematopoietic stem cell transplantation (HSCT) as soon as they have negative MRD. Patients with negative MRD may continue chemotherapy until a suitable donor is found. After completion of study treatment, patients are followed up every 4 months for 1 year, every 6 months for 1 year, and then yearly for up to 10 years.