Treatment With Amivantamab and Hyaluronidase or Cetuximab for Advanced Skin Cancer in People With a Weakened Immune System

Inclusion Criteria: * Participants must have pathologically proven diagnosis of cutaneous squamous cell carcinoma based on pathology from original diagnosis or from a metastatic/recurrent lesion * Participants must have measurable or non-measurable disease per RECIST 1.1 and must have their disease assessed by CT of chest/abdomen/pelvis (with contrast unless contraindicated) within 28 days prior to registration for measurable disease or within 42 days prior to registration for non-measurable disease. All known sites of disease must be assessed and documented on the Baseline Tumor Assessment Form (RECIST 1.1). Any lesions assessed using a non-diagnostic positron emission tomography (PET)/CT of chest/abdomen/pelvis will be considered non-measurable lesions. Pleural effusions, ascites and laboratory parameters are not acceptable as the only evidence of disease. Participants whose only measurable disease is within a previous radiation therapy port must demonstrate clearly progressive disease (in the opinion of the treating investigator) prior to registration to be considered measurable * NOTE: All diseases must be assessed and documented on the baseline tumor assessment form * Participants with exclusively locally recurrent disease must have either a contraindication to surgical treatment of lesions (i.e., complete resection is not possible or not expected to be clinically beneficial or resection conferring significant cosmetic or functional concerns) or have refused surgical or radiation treatment * Participants must be immunocompromised, defined as below. For cases where there is a lack of clarity, it is highly recommended study teams reach out to Drs. Swiecicki and Geiger for discussion: * An active diagnosis of either chronic lymphocytic leukemia (CLL) or acute leukemia, regardless of whether actively receiving therapy OR * A diagnosis of lymphoma or multiple myeloma either on antineoplastic therapy, or within 6 months after therapy completion OR * Recipient of an organ transplant (excluding corneal transplants or lung transplants) * If a transplant patient, documentation from the patient's transplant physician confirming that the patient's allograft is stable. Documentation must be dated within 180 days of registration OR * Autoimmune disease under active treatment with an immunosuppressive medication (as defined below) * Autoimmune diseases include but are not limited to: systemic lupus erythematosus, rheumatoid arthritis, inflammatory bowel disease, vasculitis, myasthenia gravis, Guillain-Barre syndrome, autoimmune hepatitis, scleroderma, primary biliary cirrhosis, pemphigus, and bullous pemphigoid * Vitiligo, psoriasis, type 1 diabetes mellitus, hypothyroidism, or resolved childhood asthma/atopy are not eligible diagnoses * Immunosuppressant medications include the following: * Tumor necrosis factor (TNF) inhibitors (adalimumab, certolizumab, etanercept, golimumab, and infliximab) * Interleukin inhibitors (anakinra, ustekinumab, secukinumab, sarilumab, siltuximab, sulfasalazine, tildrakizumab, tocilizumab, chloroquine, and hydroxychloroquine) * Janus kinase (JAK) inhibitors (baricitinib, filgotinib, and tofacitinib) * Calcineurin inhibitors (cyclosporine and tacrolimus) * Metabolic inhibitors (azathioprine, leflunomide, mercaptopurine, methotrexate) * mTOR (mammalian target of rapamycin) inhibitors (sirolimus \[rapamycin\], everolimus, and zotarolimus) * Inosine monophosphate dehydrogenase inhibitors (mycophenolate) * Phosphodiesterase inhibitors (apremilast) * B cell inhibitors (rituximab) * T cell inhibitors (abatacept) * Glucocorticoids * Active treatment is defined as current use of any one or more of the following: * Oral glucocorticoid therapy (Prednisone equivalent \> 10 mg/day) for 30 days prior to registration * Oral or subcutaneous immunosuppressive therapy for 90 days or more prior to registration * Two or more doses of intravenous non corticosteroid immunosuppressant within 90 days prior to registration * Participants with treated brain metastases must show no evidence of progression on follow-up brain imaging after central nervous system (CNS)-directed therapy * Participants with new or progressive brain metastases (active brain metastases) or leptomeningeal disease must not require immediate CNS specific treatment at the time of study registration or anticipated during the first cycle of therapy * Participants must not have had prior treatment with cetuximab or another EGFR inhibitor within the last 365 days * Participant must be ≥ 18 years old at the time of registration * Participant must have Zubrod Performance Status of 0-2 * Participant must have a complete medical history and physical exam within 28 days prior to registration * Leukocytes ≥ 3 x 10\^3/uL (within 14 days prior to registration) * Absolute neutrophil count ≥ 1.5 x 10\^3/uL (within 14 days prior to registration) * Platelets ≥ 100 x 10\^3/uL (within 14 days prior to registration) * Total bilirubin ≤ 1.5 x institutional upper limit of normal (ULN) unless history of Gilbert's disease. Participants with history of Gilbert's disease must have total bilirubin ≤ 5 x institutional ULN (within 14 days prior to registration) * Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) ≤ 3 × institutional ULN with the exception of subjects with documented liver metastases: AST and/or ALT ≤ 5.0 x institutional ULN (within 14 days prior to registration) * Participants must have a measured OR calculated creatinine clearance ≥ 30 mL/min using the following Cockcroft-Gault Formula. This specimen must have been drawn and processed within 14 days prior to registration * Participants must not have an active or past medical history of interstitial lung disease (ILD)/pneumonitis, including drug-induced or radiation ILD/pneumonitis * Participants must not have a history of lung transplantation * Participants must not have a history of pulmonary graft versus host disease (GVHD) * Participants must have adequate cardiac function. Participants with known history or current symptoms of cardiac disease, must have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, participants must be class 2B or better * Participants with a history human immunodeficiency virus (HIV)-infection must be on effective anti-retroviral therapy at registration and have undetectable viral load test on the most recent test results obtained within 6 months prior to registration * Participants with a history of chronic hepatitis B virus (HBV) infection must have undetectable HBV viral load while on suppressive therapy on the most recent test results obtained within 6 months prior to registration, if indicated * Participants with a history of hepatitis C virus (HCV) infection must have been treated and cured. Participants currently being treated for HCV infection must have undetectable HCV viral load test on the most recent test results obtained within 6 months prior to registration, if indicated * Participants must not have an uncontrolled illness, including but not limited to: * Ongoing or active infection (includes infection requiring treatment with antimicrobial therapy \[participants will be required to complete antibiotics 1 week prior to starting study treatment\]) * Active bleeding diathesis * Any ophthalmologic condition that is clinically unstable * Participants must not have a prior or concurrent malignancy whose natural history or treatment (in the opinion of the treating physician) has the potential to interfere with the safety or efficacy assessment of the investigational regimen * Participants must not be pregnant or nursing (nursing includes breast milk fed to an infant by any means, including from the breast, milk expressed by hand, or pumped) due to known toxicities of amivantamab. Individuals who are of reproductive potential must have agreed to use an effective contraceptive method with details provided as a part of the consent process. A person who has had menses at any time in the preceding 12 consecutive months or who has semen likely to contain sperm is considered to be of "reproductive potential." In addition to routine contraceptive methods, "effective contraception" also includes refraining from sexual activity that might result in pregnancy and surgery intended to prevent pregnancy (or with a side-effect of pregnancy prevention) including hysterectomy, bilateral oophorectomy, bilateral tubal ligation/occlusion, and vasectomy with testing showing no sperm in the semen. Participants must agree not to donate ova or sperm for the purpose of reproduction during the study and for a minimum of 6 months after receiving the last dose of study treatment. For female participants of childbearing potential, a negative pregnancy test is required within 72 hours prior to registration * Participants must be offered the opportunity to participate in specimen banking