This phase I/II trial studies the side effects and best dose of recombinant interleukin-12 when given together with cetuximab and to see how well they work in treating patients with squamous cell carcinoma of the head and neck that has come back, spread to another place in the body, or cannot be removed by surgery. Recombinant interleukin-12 may stimulate the white blood cells to kill tumor cells. Monoclonal antibodies, such as cetuximab, may interfere with the ability of tumor cells to grow and spread. Giving recombinant interleukin-12 together with cetuximab may kill more tumor cells.