ABemacicliB or Abemaciclib and HydroxYchloroquine to Target Minimal Residual Disease in Breast Cancer

Inclusion Criteria: * Histologically-confirmed, primary, invasive breast cancer diagnosed within 5 years of entry into the companion DTC screening protocol UPCC 28115 * Qualifying risk status, at diagnosis utilizing receptor testing by ASCO/CAP guidelines, meting at least one of the following: (i) Lymph node involvement a. Pathologically-confirmed positive axillary lymph nodes, regardless of receptors, or (b) Evidence of lymph node involvement by preoperative imaging AND fibrosis and/or treatment effect in lymph nodes after neoadjuvant therapy, regardless of receptor status (ii) Primary tumor that is ER/PR/Her2 negative: estrogen receptor (ER) \< 10%, progesterone receptor (PR) \< 10% and negative for Her2-overexpression by ASCO-CAP guidelines, regardless of lymph nodes status (iii) Primary tumor that is ER+/Her2 negative/Lymph node negative with Breast Cancer Recurrence Score of \>/= 25 per the Genomic Health Oncotype DX breast cancer test and/or high risk MammaPrint (iv) Evidence of residual disease in the breast on pathologic assessment after neoadjuvant chemotherapy * Patients must have completed all primary therapy (definitive surgery, (neo)adjuvant chemotherapy adjuvant radiation and/or Her2-directed therapy) for the index malignancy at least 4 weeks prior to study entry. Prior treatment-related toxicity must be resolved or improving to Grade 1 with the exception of alopecia, Grade 2 endocrine disorders (e.g. adrenal insufficiency or thyroid disorders from prior immunotherapy) controlled on stable replacement therapy for \> 1 year, and up to Grade 3 peripheral neuropathy, prior to study enrollment. Concurrent receipt of adjuvant endocrine and bone modifying agents is allowed per standard of care guidelines. Tamoxifen is not allowed due to drug-drug interactions with HCQ. * Bone marrow aspirate obtained via research trial UPCC 28115 after completion of therapy (except endocrine therapy) demonstrates detectable DTCs (via IHC) * No evidence of recurrent local or distant breast cancer by physical examination, blood tests (CBC, LFTs, Alk Phos), or imaging. Assessment for overt metastatic disease by radiologic testing per institutional guidelines (CT Chest, Abdomen and Pelvis, bone scan, MRI and/or PET/CT) will only be done in patients with DTCs detected on bone marrow aspirate who are being screened for this trial. * Age \>/= 18 years * ECOG performance status =/\< 2 * Ability to swallow oral medications * No contraindications to the study medications or uncontrolled medical illness. * Adequate bone marrow function as shown by: ANC \>/= 1.5 x 10\^9/L, Platelets \>/= 100 x 10\^9/L, Hb \>9 g/dL * Adequate liver function as shown by: Serum bilirubin \</= 1.5 x ULN, ALT and AST \</= 3.0 x ULN, and INR \</=1.5 * Adequate renal function: serum creatinine \</= 1.5 x ULN * Adequate muscle function: creatinine phosphokinase (CPK) \</= 2.5 x ULN * Anticoagulation is allowed if target INR =/\< 1.5 on a stable dose of warfarin or on a stable dose of anticoagulant for \>2 weeks at time of randomization * Ability to provide informed consent Exclusion Criteria * Concurrent enrollment on another investigational therapy * Patient has received an experimental treatment in a clinical trial within the last 30 days or 5 half-lives, whichever is longer, prior to randomization (or treatment assignment), or is currently enrolled in any other type of medical research (for example: medical device) judged by the sponsor not to be scientifically or medically compatible with this study. * Prior treatment with abemaciclib is not allowed. (Patients may have received prior CDK4/6 inhibitor therapy with an agent other than abemaciclib. Patients must have discontinued CDK4/6 inhibitor at least 6 months prior to screening.) * Known hypersensitivity to hydroxychloroquine or any of its derivatives * Prior hydroxychloroquine exposure for a duration of \> 1 month since the completion of the patient's primary therapy (definitive surgery, (neo)adjuvant chemotherapy, adjuvant radiation, and/or Her2-directed therapy) for the index malignancy. * Patients with hormone-receptor positive breast cancer may not be receiving tamoxifen due to drug-drug interactions with hydroxychloroquine * Patients who have initiated bone modifying agents within 3 months prior to the start of study treatment * Patients who have had major surgery within 14 days prior to randomization (or treatment assignment) * Patients who have any severe and/or uncontrolled medical conditions or other conditions that could affect their participation in the study * Female patients who are pregnant or breast feeding, or adults of reproductive potential who are not using effective birth control methods. * Women of childbearing potential must have a negative serum pregnancy test within 7 days prior to the first dose of Abema