Individuals with FA have a higher risk of developing head and neck squamous cell carcinoma (HSNCC) at young ages. Long-term survival rates remain low because often these patients cannot tolerate conventional chemotherapy and radiation treatments due to issues related to toxicity. This study is the first clinical trial focused on a therapy for FA-related HNSCC. It will investigate the efficacy and safety of the drug, Afatinib, when administered to people with FA who are diagnosed with unresectable and/or advanced squamous cell carcinoma of the oral cavity, oropharynx, hypopharynx, or larynx.