Inclusion Criteria:

* Patients must meet all SCREENING/PRE-SCREENING and SUB-STUDY REGISTRATION COMMON ELIGIBILITY CRITERIA as specified in S1400: Phase II/III Biomarker-Driven Master Protocol for Previously Treated Squamous Cell Lung Cancer (Lung-Map)
* Patients must have been assigned to S1400A
* Patients must not have any prior exposure to immunotherapy such as, but not limited to anti-programmed death 1 (PD-1) or anti-PD-L1 antibodies; prior exposure to the following is allowed: anti-cytotoxic T lymphocyte antigen 4 (CTLA-4) antibodies, live attenuated vaccines, anti-EGFR agents and sargramostim (GM-GSF)
* Patients must not have received nitrosoureas or mitomycin-C within 42 days prior to sub-study registration
* Patients must not have any active or prior documented autoimmune or inflammatory disease (including inflammatory bowel disease, diverticulitis with the exception of diverticulosis, celiac disease, irritable bowel disease; Wegener syndrome; Hashimoto syndrome) within 3 years prior to sub-study registration; patients with vitiligo, alopecia, Grave's disease, or psoriasis requiring systemic treatment within the past 3 years are not eligible
* Patients must not have any history of primary immunodeficiency
* Patients must not have received any immunosuppressive medication within 28 days prior to sub-study registration and must not be planning to receive any such agents while on protocol treatment; however, intranasal and inhaled corticosteroids or systemic corticosteroids at physiologic doses not to exceed 10 mg/day of prednisone or equivalent are allowed
* Patients must not have any prior grade >= 3 immune-related adverse event (irAE) or any unresolved irAE > grade 1
* Patients must not have any history of organ transplant that requires use of immunosuppressives
* Patients must not have any known allergy or reaction to any component of the MEDI4736 formulation
* Patients must not have a known history of tuberculosis
* Patients must not have received a live attenuated vaccination within 28 days prior to sub-study registration
* Patients must not have known human immunodeficiency virus (HIV), hepatitis B or C positivity
* Patients must also be offered participation in banking for future use of specimens
* STEP 2 TO MEDI4736 RE-TREATMENT REGISTRATION:
* Patient must have progressed following 12 months of treatment with MEDI4736; patients who discontinue MEDI4736 prior to the completion of 12 months (for any reason) are not eligible; patients who have already completed two 12-month periods of treatment are not eligible
* Patients may have measurable or non-measurable disease documented by computed tomography (CT) or magnetic resonance imaging (MRI); the CT from a combined positron emission tomography (PET)/CT may be used to document only non-measurable disease unless it is of diagnostic quality; measurable disease must be assessed within 28 days prior to re-treatment registration; pleural effusions, ascites and laboratory parameters are not acceptable as the only evidence of disease; non-measurable disease must be assessed within 42 days prior to re-treatment registration; all disease must be assessed and documented on the Baseline Tumor Assessment Form; patients whose only measurable disease is within a previous radiation therapy port must demonstrate clearly progressive disease (in the opinion of the treating investigator) prior to RE-TREATMENT registration
* Patients must have a CT or MRI scan of the brain to evaluate for central nervous system (CNS) disease within 42 days prior to RE-TREATMENT registration; patient must not have leptomeningeal disease, spinal cord compression or brain metastases unless: metastases have been locally treated and have remained clinically controlled and asymptomatic for at least 14 days following treatment and prior to RE-TREATMENT registration, AND patient has no residual neurological dysfunction and has been off corticosteroids for at least 24 hours prior to RE-TREATMENT registration
* Patients must not have received any treatment after discontinuing MEDI4736 with the following exceptions; localized palliative radiation therapy is allowed for symptom management, provided and treatment is completed >= 14 days prior to RE-TREATMENT registration; local treatment for brain metastases is allowed
* Patients must not have received any immunosuppressive medication within 28 days prior to RE-TREATMENT registration and must not be planning to receive any such agents while on protocol treatment; however, intranasal and inhaled corticosteroids or systemic corticosteroids at physiologic doses not to exceed 10 mg/day of prednisone or equivalent are allowed
* Patients must not have any prior grade >= 3 immune-related adverse event (irAE) or any unresolved irAE > grade 1 (MEDI4736 RE-TREATMENT)
* Patients must not have received a live attenuated vaccination within 28 days prior to RE-TREATMENT registration
* Patients must not have known HIV, hepatitis B or hepatitis C positivity
* Patients must not be planning to receive any concurrent chemotherapy, immunotherapy, biologic or hormonal therapy for cancer treatment; concurrent use of hormones for non-cancer-related conditions (e.g., insulin for diabetes and hormone replacement therapy) is acceptable
* Absolute neutrophil count (ANC) >= 1,500/mcl obtained within 28 days prior to RE-TREATMENT registration
* Platelet count >= 100,000 mcl obtained within 28 days prior to RE-TREATMENT registration
* Hemoglobin >= 9 g/dL obtained within 28 days prior to RE-TREATMENT registration
* Serum bilirubin =< institutional upper limit of normal (IULN) within 28 days prior to RE-TREATMENT registration; for patients with liver metastases, bilirubin must be =< 5 x IULN
* Either alanine aminotransferase (ALT) or aspartate aminotransferase (AST) =< 2 x IULN within 28 days prior to RE-TREATMENT registration (if both ALT and AST are done, both must be =< 2 IULN); for patients with liver metastases, either ALT or AST must be =< 5 x IULN (if both ALT and AST are done, both must be =< 5 x IULN)
* Patients must have a serum creatinine =< the IULN OR measured or calculated creatinine clearance >= 50 mL/min using the Cockcroft-Gault formula
* Patients must have Zubrod performance status of 0-1 documented within 28 days prior to RE-TREATMENT registration
* Prestudy history and physical exam must be obtained within 28 days prior to RE-TREATMENT registration
* Patients must not be pregnant or nursing; women/men of reproductive potential must have agreed to use an effective contraceptive method; a woman is considered to be of "reproductive potential" if she has had menses at any time in the preceding 12 consecutive months; in addition to routine contraceptive methods, "effective contraception" also includes heterosexual celibacy and surgery intended to prevent pregnancy (or with a side-effect of pregnancy prevention) defined as a hysterectomy, bilateral oophorectomy or bilateral tubal ligation; however, if at any point a previously celibate patient chooses to become heterosexually active during the time period for use of contraceptive measures outlined in the protocol, he/she is responsible for beginning contraceptive measures
* As a part of the Oncology Patient Enrollment Network (OPEN) registration process the treating institution's identity is provided in order to ensure that the current (within 365 days) date of institutional review board approval for this study has been entered in the system
* Patients with impaired decision-making capacity are eligible as long as their neurological or psychological condition does not preclude their safe participation in the study (e.g., tracking pill consumption and reporting adverse events to the investigator)
* Patients must be informed of the investigational nature of this study and must sign and give written informed consent in accordance with institutional and federal guidelines