This phase III trial compares the effect of adding cemiplimab to standard therapy (surgery with or without radiation) versus standard therapy alone in treating patients with stage III/IV squamous cell skin cancer that is able to be removed by surgery (resectable) and that may have come back after a period of improvement (recurrent). The usual treatment for patients with resectable squamous cell skin cancer is the removal of the cancerous tissue (surgery) with or without radiation, which uses high energy x-rays, particles, or radioactive seeds to kill cancer cells and shrink tumors. Immunotherapy with monoclonal antibodies, such as cemiplimab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. Cemiplimab has been approved for the treatment of skin cancer that has spread or that cannot be removed by surgery, but it has not been approved for the treatment of skin cancer than can be removed by surgery. Adding cemiplimab to the usual treatment of surgery with or without radiation may be more effective in treating patients with stage III/IV resectable squamous cell skin cancer than the usual treatment alone.