Targeted T Cells After Neoadjuvant Chemotherapy in Treating Women With Stage II or III Breast Cancer Undergoing Surgery

Inclusion Criteria * Signed and dated institutional review board (IRB)-approved consent form * Women of reproductive potential must agree to use an effective nonhormonal method of contraception during therapy * Eastern Cooperative Oncology Group (ECOG) performance status (PS) of 0 or 1 and/or Karnofsky PS of \>= 70% * Diagnosis of invasive adenocarcinoma of the breast made by core needle biopsy * Palpable primary breast tumor measuring \>= 2.0 cm on physical exam or imaging prior to neoadjuvant chemotherapy * Patients with stage II-IIIB breast cancer that is HER2-negative by immunohistochemistry (IHC) (0-2+) and fluorescence in situ hybridization (FISH) (HER2/chromosome enumeration probe \[CEP\]17 amplification ratio \< 2.0) who have completed "third generation" neoadjuvant chemoT and planned local treatment (surgery and radiation if indicated); estrogen receptor (ER) or progesterone receptor (PR) status should be negative * Patients may have lymph node positive or negative disease, as long as they have clinical/pathologic stage II or IIIB breast cancer; patients may have the lymph nodes assessed by any method deemed appropriate by the treating physicians, including pre-neoadjuvant therapy sentinel lymph node biopsy * Presence of residual disease measuring at least 5mm (as single foci or in aggregate) on final pathology following surgery * Patients must discontinue sex hormone therapy prior to registration, e.g. birth control pills, hormonal replacement therapy * Absolute neutrophil count (ANC) must be \>= 1000/mm\^3 * Platelet count must be \>= 100,000/mm\^3 * Hemoglobin must be \>= 9.0 mg/dL * Total bilirubin must be =\< the upper limit of normal (ULN) for the lab unless the patient has a grade 1 bilirubin elevation (\> ULN to 1.5 x ULN) resulting from Gilbert's disease or similar syndrome due to slow conjugation of bilirubin; and * Alkaline phosphatase must be =\< 2.5 x ULN for the lab * Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) must be =\< 1.5 x ULN for the lab * Alkaline phosphatase and AST/ALT may not both be \> the ULN; for example, if the alkaline phosphatase is \> the ULN but =\< 2.5 x ULN, then the AST/ALT must be =\< the ULN; if the AST/ALT is \> the ULN but =\< 1.5 x ULN, then the alkaline phosphatase must be =\< ULN * Patients with either skeletal pain or alkaline phosphatase that is \> ULN must have a bone scan showing they do not have metastatic disease; suspicious findings on bone scan must be confirmed as benign by x-ray, magnetic resonance imaging (MRI), or biopsy * Patients with AST/ALT or alkaline phosphatase \> ULN must have liver imaging that does not demonstrate metastatic disease * Patients with AST/ALT \> ULN must have negative hepatitis studies * Patients with stage II disease and clinical suspicion for metastatic disease based on reported symptoms, physical examination findings, or laboratory abnormalities must have staging studies demonstrating no evidence of metastatic disease (with exception of axillary lymph nodes or mammary nodes); patients with stage IIIA disease must have staging studies demonstrating no evidence of metastatic disease (with exception of axillary lymph nodes or mammary nodes), even if asymptomatic with normal physical examination and laboratory values; such staging studies must include: chest imaging (chest X-ray, computed tomography \[CT\], or MRI), abdominal/pelvis imaging (CT or MRI), and bone imaging (bone scan or positron emission tomography \[PET\]-scan); abnormalities that are indeterminate and too small to biopsy should be followed with further imaging, as appropriate, but do not exclude patients from the study; abnormalities that are suspicious and large enough to biopsy exclude patients from the study, unless a biopsy is performed and is negative for metastatic disease * Serum creatinine =\< 1.5 x ULN for the lab * Pre-entry core biopsy with sufficient material for correlative studies * Left Ventricular Ejection Fraction (LVEF) \>= 45 % (by multigated acquisition scan \[MUGA\] or echocardiography) Exclusion Criteria * Tumor determined to be HER2-positive by immunohistochemistry (3+) or by fluorescent in situ hybridization (HER2/CEP17 amplification ratio \>= 2.0) * Tumors clinically staged as anyT with N3 disease or unresectable disease * Evidence of disease progression on neoadjuvant chemo T * Definitive evidence of metastatic disease with exception of axillary lymph nodes or mammary nodes * Synchronous bilateral breast cancer (invasive or ductal carcinoma in situ \[DCIS\]) * Treatment with biotherapy, and/or hormonal therapy for the currently diagnosed breast cancer prior to study entry * Any sex hormonal therapy, e.g., birth control pills, ovarian hormonal replacement therapy, etc. within 2 weeks prior to the collection of cells * Prior history of invasive breast cancer (patients with a history of DCIS or lobular carcinoma in situ \[LCIS\] are eligible) * Other malignancies unless the patient is considered to be disease-free for 5 or more years prior to randomization and is deemed by the physician to be at low risk for recurrence; patients with the following cancers are eligible if diagnosed and treated within the past 5 years: carcinoma in situ of the cervix, carcinoma in situ of the colon, melanoma in situ, and basal cell or squamous cell carcinoma of the skin * Known cardiac disease which precludes their ability to receive planned treatments: * Angina pectoris that requires the use of anti-anginal medication * History of documented congestive heart failure * Serious cardiac arrhythmia requiring medication * Severe conduction abnormality * Valvular disease with documented cardiac function compromise; and * Uncontrolled hypertension defined as blood pressure (BP) that is consistently \> 150/90 on antihypertensive therapy at the time of registration (Patients with hypertension that is well controlled on medication are eligible) * History of myocardial infarction (MI) documented by elevated cardiac enzymes with persistent regional wall motion abnormality on assessment of left ventricular (LV) function (patients with history of MI must have an echo instead of/in addition to a MUGA to evaluate LV wall motion) * Symptomatic peripheral vascular disease * Other non-malignant systemic disease (cardiovascular, renal, hepatic, etc.) that would preclude treatment with any of the treatment regimens or would prevent required follow-up * Chronic ongoing oral steroid use at the time of registration for any condition (such as asthma, rheumatoid arthritis, etc) * Administration of any investigational agents within 30 days before study entry * Pregnancy or lactation at the time of registration * Psychiatric or addictive disorders or other conditions that in the opinion of the investigators would preclude the patient from complying with the study protocol