Gefitinib in Treating Patients Who Are Undergoing Surgery and/or Radiation Therapy for Locally Advanced or Recurrent Squamous Cell Skin Cancer

Inclusion Criteria: * Within 12 weeks (+/- 2 weeks) prior to study entry, patients must have histologically or cytologically confirmed squamous cell carcinoma (SCC) of skin that is either locally advanced or recurrent with measurable disease; if the biopsy was collected outside of MDACC, the MDACC Pathology Department must assess and confirm the SCC diagnosis * Patients may have previous surgical intervention with residual or recurrent disease * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) * Leukocytes \>= 3,000/mm\^3 * Absolute neutrophil count \>= 1,500/mm\^3 * Platelets \>= 100,000/mm\*3 * Total bilirubin within normal institutional limits * aspartate aminotransferase (AST or SGOT) and alanine aminotransferase (ALT or SGPT) =\< 2.5 \* institutional 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 * Tumors must be at least 2 cms in size or have histological or cytological verification of muscle, bone, lymph node metastasis, or perineural involvement, as measured by the treating physician(s) or National principal investigator (PI) * Negative serum pregnancy test for women of child-bearing potential (performed within 14 days, +/- 1 day, prior to start of treatment); women of child-bearing potential and men must agree to use adequate contraception prior to study entry and for the duration of study participation; should a woman become pregnant or suspect she is pregnant while participating in the study, she should inform her treating physician(s) immediately * Ability to understand and the willingness to sign a written Informed Consent Document (ICD); in the event that non-English speaking participants are eligible for this study, a short form (if applicable) or an ICD in their language, will be utilized and completed in accordance with the MD Anderson's "Policy For Consenting Non-English Speaking Participants" Exclusion Criteria: * Patients who have previous radiotherapy to the proposed site of skin cancer * Patients with active cancers other than skin * Patients currently receiving any other investigational agents at time of study enrollment; patients may have received investigational agents in the past; no washout time period is required * Patients with a history of brain metastases must be excluded from this clinical study because of their poor prognosis and because they often develop progressive neurological dysfunction that would confound the evaluation of neurological and other adverse events * History of allergic reactions attributed to compounds of similar chemical or biologic composition to ZD1839 * Age less than 18 years * Presence of uncontrolled intercurrent illness (co-morbid conditions) that would limit compliance with study requirements including , but not limited to, ongoing or active infection requiring parenteral antibiotics at time of study registration, symptomatic congestive heart failure (NYHA class II or greater), unstable angina pectoris or cardiac arrhythmia requiring maintenance medication * Pregnant women are excluded from this study because ZD1839 is a signal transduction inhibitor agent with the potential for teratogenic or abortifacient effects; there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with ZD1839, breastfeeding should be discontinued if the mother is treated with ZD1839 * Patients with known immune deficiency are at an increased risk when treated with marrow-suppressive therapy, HIV-positive patients receiving combination anti-retroviral therapy are excluded due to the possible pharmacokinetic interactions with ZD1839; appropriate studies will be undertaken in patients receiving combination anti-retroviral therapy when indicated * CYP3A4 inducing agents; patients receiving the following CYP3A4 inducing agents will be excluded; these include: carbamazepine, ethosuximide, griseofulvin, modafinil, nafcillin, oxcarbazepine, Phenobarbital, phenylbutazone, phenytoin, rifampin, rifabutin, St. John's Wort, and sulfinpyrazone * Patients with distant metastatic disease as determined by diagnostic imaging (i.e., chest x-rays) and/or hematologic assessments (i.e., liver enzymes)