This randomized phase III trial compares nivolumab with ipilimumab and nivolumab alone in treating patients with stage IV squamous cell lung cancer that has come back after previous treatment. This is a "non-match" sub-study that includes all screened patients not eligible for a biomarker-driven sub-study. Monoclonal antibodies, such as nivolumab and ipilimumab, may be able to shrink tumors. It is not yet known whether nivolumab works better with or without ipilimumab in treating patients with squamous cell lung cancer.