Erlotinib in Treating Patients With Recurrent or Metastatic Skin Squamous Cell Carcinoma

Inclusion Criteria: * Have histologically or cytologically confirmed cutaneous squamous cell carcinoma (CSCC) that is not amenable to curative therapy. If the biopsy was collected outside of MD Anderson Cancer Center (MDACC), the MDACC Pathology Department must assess and confirm the squamous cell carcinoma (SCC) diagnosis. * Have measurable disease. * Have Eastern Cooperative Oncology Group (ECOG) performance status 0-2. * Must have 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 MDACC "Policy For Consenting Non-English Speaking Participants. * Leukocytes \>= 3,000/mm\^3. * Absolute neutrophil count \>= 1,500/mm\^3. * Platelets \>= 75,000/mm\^3. * Hemoglobin \>= 8g/dL. * Total bilirubin =\< 2 x institutional upper limit of normal (ULN). * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (serum glutamic pyruvic transaminase \[SGPT\]) =\< 2.5 x ULN if alkaline phosphatase is normal, or alkaline phosphatase =\< 4 x ULN if transaminases are normal. * Creatinine =\< 2.0 x ULN or creatinine clearance \>= 60 mL/min/1.73 m\^2. * Prior radiotherapy is allowed if: (a) there is measurable disease outside the radiation field OR (b) radiotherapy was completed more than 4 weeks ago and there is clearly recurrent and growing disease within the radiation field. * Must be able to take intact tablets by mouth, or be able to take tablets dissolved in water by mouth or by a percutaneous gastrostomy tube. * Patients - both males and females - with reproductive potential (includes women who are menopausal for less than 1 year and not surgically sterilized) must practice effective contraceptive measures such as barrier methods, condom or diaphragm with spermicide, or abstinence throughout the study. Birth control should continue for 4 weeks after discontinuation of erlotinib therapy. Women of childbearing potential must provide a negative pregnancy test (serum beta human chorionic gonadotropin \[HCG\]) within 72 hours prior to first receiving protocol therapy. * Organ transplant patients are eligible as long as they do not have active signs of rejection and have adequate bone marrow function. Exclusion Criteria: * Women who are pregnant, breastfeeding, and women and men not practicing effective birth control. Erlotinib 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 erlotinib. Breastfeeding should be discontinued if the mother is treated with erlotinib. * Prior estimated glomerular filtration rate (EGFR) inhibitor therapy is not allowed (including, but not limited to, erlotinib, gefitinib, cetuximab, panitumumab, vandetanib). * Patients who are receiving any other anticancer or investigational agents at time of study enrollment. Patients may have received one other systemic therapy or investigational agent in the past, but a washout time period of at least 4 weeks and recovery of any treatment-related toxicities to \< Common Terminology Criteria for Adverse Events version 4 (CTCAEv4) grade 2 is required. * History of allergic reactions attributed to compounds of similar chemical or biologic composition to erlotinib. * Patients with a history of an invasive malignancy (other than the one treated in this study) or lymphoproliferative disorder within the past 3 years. Patients with a history of adequately treated non-melanoma skin cancer, ductal carcinoma in situ of the breast, or carcinoma in situ of the cervix are allowed. * Patients with incomplete healing from previous surgery. * Patients with pulmonary fibrosis (other than in a radiated field) or active interstitial lung disease. * Patients with active gastrointestinal disease or a disorder that alters gastrointestinal motility or absorption, including lack of integrity of the gastrointestinal tract (for example, a significant surgical resection of the stomach or small bowel, inflammatory bowel disease or uncontrolled chronic diarrhea. * Patients with skin rash CTCAEv4 grade 2. * In the opinion of the investigator, patients with any condition that is unstable or could jeopardize the safety of the patient or could limit compliance with the study's requirements. These include, but are not limited to, ongoing or active infection requiring parenteral antibiotics at time of study registration, psychiatric illness that would limit compliance with study requirements or symptomatic congestive heart failure (New York Heart Association \[NYHA\] class II or greater), unstable angina pectoris or cardiac arrhythmia requiring maintenance medication. * Patient is unwilling or unable to discontinue prohibited concomitant therapies, (i.e St. John's wort, grapefruit juice, histamine type 2 receptor \[H2\] blockers/proton pump inhibitors, strong CYP3A4 inhibitors and inducers).