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 or MRI 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 diagnosis of either chronic lymphocytic leukemia (CLL), acute leukemia, myelodysplastic syndrome, polycythemia vera, or myelofibrosis 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 * Participants must have Zubrod Performance Status of 0-2 * Participants 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 * Pa