Lung-MAP: Nivolumab With or Without Ipilimumab as Second-Line Therapy in Treating Patients With Recurrent Stage IV Squamous Cell Lung Cancer and No Matching Biomarkers

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 S1400I * Patients must not have had prior treatment with an anti-programmed cell death (PD)-1, anti-PD-L1, anti-PD-L2, anti-cytotoxic T-lymphocyte-associated protein (CTLA)-4 antibody, or any other antibody or drug specifically targeting T-cell costimulation or immune checkpoint pathways * Patients must not have an active, known, or suspected autoimmune disease; patients are permitted to enroll if they have vitiligo, type I diabetes mellitus, hypothyroidism only requiring hormone replacement, psoriasis not requiring systemic treatment, or conditions not expected to recur in the absence of an external trigger * Patients must not have any known allergy or reaction to any component of the nivolumab and ipilimumab formulations * Patients must not have received systemic treatment with corticosteroids (\> 10 mg daily prednisone or equivalent) or other immunosuppressive medications within 14 days prior to sub-study registration; inhaled or topical steroids, and adrenal replacement doses =\< 10 mg daily prednisone or equivalent are permitted in the absence of active autoimmune disease * Patients must not have a known positive test for hepatitis B virus surface antigen (HBV sAg) or hepatitis C virus ribonucleic acid (HCV antibody) indicating acute or chronic infection; patients with a positive hepatitis C antibody with a negative viral load are allowed * Patients must not have known history of testing positive for human immunodeficiency virus (HIV) or known acquired immunodeficiency syndrome (AIDS) * Patients must not have interstitial lung disease that is symptomatic or disease that may interfere with the detection or management of suspected drug-related pulmonary toxicity * Patients must also be offered participation in banking for future use of specimens * Patients must have a lipase, amylase, TSH with reflex free T3/T4 performed within 7 days prior to sub-study registration * Patients must not have any grade III/IV cardiac disease as defined by the New York Heart Association Criteria (i.e., patients with cardiac disease resulting in marked limitation of physical activity or resulting in inability to carry on any physical activity without discomfort), unstable angina pectoris, and myocardial infarction within 6 months, or serious uncontrolled cardiac arrhythmia * Patients with a history of congestive heart failure (CHF) or at risk because of underlying cardiovascular disease or exposure to cardiotoxic drug should have an electrocardiogram (EKG) and echocardiogram performed to evaluate cardiac function as clinically indicated * Patients with evidence of congestive heart failure (CHF), myocardial infarction (MI), cardiomyopathy, or myositis should have a cardiac evaluation including lab tests and cardiology consultations as clinically indicated including EKG, creatine phosphokinase (CPK), troponin, and echocardiogram * Patients who can complete Patient Reported Outcomes (PRO) forms in English are required to complete a pre-study S1400I Patient Reported Outcomes (PRO) Questionnaire and a pre-study S1400I European Quality of Life Five Dimension (EQ-5D) Questionnaire within 14 days prior to registration; NOTE: Patients enrolled to S1400I prior to 9/1/2016 are not eligible for the PRO study