This randomized phase II trial studies how well erlotinib hydrochloride or crizotinib with chemoradiation therapy works in treating patients with stage III non-small cell lung cancer. Radiation therapy uses high energy x rays to kill tumor cells. Specialized radiation therapy that delivers a high dose of radiation directly to the tumor may kill more tumor cells and cause less damage to normal tissue. Drugs used in chemotherapy, such as cisplatin, etoposide, paclitaxel, and carboplatin, work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. It is not yet known whether giving erlotinib hydrochloride is more effective than crizotinib with chemoradiation therapy in treating patients with non-small cell lung cancer.