Cetuximab and Recombinant Interleukin-12 in Treating Patients With Squamous Cell Carcinoma of the Head and Neck That is Recurrent, Metastatic, or Cannot Be Removed by Surgery

Inclusion Criteria: * Patients must have histologically-proven recurrent and/or metastatic squamous cell carcinoma of the head and neck that is unresectable; patients in the phase II portion of the trial must have measurable disease * Any number of prior systemic therapies for metastatic/recurrent disease are permitted in both the phase I and II portions of the study; patients need not have received a prior cetuximab-based chemotherapy regimen to be eligible for this trial * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) * Life expectancy of greater than 6 months * Leukocytes \>= 3,000/mcL * Absolute neutrophil count \>= 1,500/mcL * Platelets \>= 100,000/mcL * Total bilirubin within normal institutional limits * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 times upper limit of normal * Creatinine within normal institutional limits OR creatinine clearance \>= 60 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal * The effects of IL-12 on the developing human fetus are unknown; for this reason, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation; should a woman become pregnant or suspect she is pregnant while participating in this stu…