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 study, she should inform her treating physician immediately * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Patients who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who have not recovered from adverse events due to agents administered more than 4 weeks earlier * Patients may not be receiving any other investigational agents * Patients with known brain metastases may be enrolled if this site of disease has been adequately treated, the patient does not require steroids, and the patient has been stable for at least 3 months prior to enrollment * History of allergic reactions attributed to compounds of similar chemical or biologic composition to IL-12 or other agents used in study * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Pregnant women are excluded from this study because IL-12 is an agent with the potential for teratogenic or abortifacient effects; because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with IL-12, breastfeeding should be discontinued if the mother is treated with IL-12; these potential risks may also apply to other agents used in this study