This randomized phase II/III compares rilotumumab when given together with erlotinib hydrochloride against erlotinib hydrochloride alone in treating patients with stage IV squamous cell lung cancer that has come back after previous treatment. This is a sub-study that includes all screened patients positive for the met proto-oncogene (MET)/hepatocyte growth factor (HGF) biomarker. HGF can interact with MET and can cause tumor cells to grow more quickly. Rilotumumab may decrease the activity of HGF and may be able to shrink tumors. Erlotinib hydrochloride may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. It is not yet known whether giving rilotumumab with erlotinib hydrochloride works better than erlotinib hydrochloride alone (standard treatment) in treating squamous cell lung cancer.