The purpose of this study is to find out if the combination of two established anti-cancer therapies are beneficial in participants with Head and Neck Squamous Cell Carcinoma (HNSCC). Specifically, investigators want to determine if the combination of Cetuximab and nivolumab can help people with advanced cases of HNSCC. Both cetuximab and nivolumab have been used separately to treat HNSCC and are Food and Drug Administration (FDA) approved in this type of cancer.

PHASE I: Participants will be enrolled sequentially and treated at Dose Level 1, or Dose Level -1, every 2 weeks for 12 cycles or until discontinuation.

Each cycle is 4 weeks. Cetuximab is given alone in lead-in period at Day -14 before Cycle 1 only. In all subsequent doses starting Cycle 1 Day 1, nivolumab and cetuximab will be given concurrently. Dose limiting toxicity (DLT) assessment will be performed during Cycle 1 and will start with the initiation of the combination of cetuximab and nivolumab (4 weeks).

PHASE II: Once the maximum tolerated dose (MTD) or the recommended phase II dose of cetuximab is determined in Phase I, accrual to the phase II will begin.

FOLLOW-UP: Participants will be followed for 2 years from End of Treatment. The imaging studies will be performed every 8 weeks (2 cycles) of the treatments during Cycle 1-6 and then every 12 weeks during Cycle 7-12 as per standard of care. Patient will be followed by treating physicians as per standard of care.

Inclusion Criteria:

• Participants must have histologically or cytologically confirmed squamous cell carcinoma of oral cavity, oropharynx, paranasal sinuses, nasal cavity, hypopharynx, or larynx. Squamous cell carcinoma of unknown primary in cervical lymph node can be included only if p16 status is positive.
• Must have recurrent or metastatic HNSCC stage III/IV that is not amenable to local therapy with curative intent (surgery or radiation therapy with or without chemotherapy).Patients with persistent disease following radiation therapy administered with a chemotherapy sensitizer may also be included.
• Must have progressed on at least one prior line of chemotherapy, targeted therapy, palliative radiation, and/or biological therapy regimen for their recurrent and/or metastatic HNSCC. However, if patients are likely to be intolerant to standard first-line systemic chemotherapy, the patients are eligible to enroll to this study as the first-line therapy. Additionally, patients with persistent disease or platinum-refractory recurrent disease may enroll in this study as a first-line therapy.
• Must NOT have any systemic therapy for recurrent and/or metastatic disease except if given as a part of a multimodality treatment (i.e. re-irradiation and systemic therapy for curable intent of locally recurrent disease).
• Must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded) as outlined in RECIST version 1.1.
• Must be ≥ 18 years of age.
• Life expectancy of greater than 3 months.
• Eastern Cooperative Oncology Group (ECOG) performance status ≤ 2.
• Must have normal organ function: Absolute neutrophil count > 1,500/μL; Hemoglobin > 9 g/dL; Platelets > 100,000/μL; Total bilirubin ≤ 1.5 mg/dL X institutional upper limits of normal (ULN); AST (SGOT)/ALT (SGPT) < 3 X institutional ULN (or 5.0 X the ULN in the setting of liver metastasis); Serum creatinine of ≤ 1.5 X ULN or creatinine clearance > 40 mL/minute (using Cockcroft/Gault formula): Female creatinine clearance = (140 - age in years) x weight in kg x 0.8572 x serum creatinine in mg/ dL; Male creatinine clearance = (140 - age in years) x weight in kg x 1.0072 x serum creatinine in mg/dL.
• Participants, if sexually active, must be postmenopausal, surgically sterile, or using effective contraception (hormonal or barrier methods). Female participants of childbearing potential must have a negative serum pregnancy test within 7 days prior to enrollment.
• Ability to understand and the willingness to sign a written informed consent document.

Exclusion Criteria:

• Have experienced grade 3 or above skin toxicity from prior Epidermal growth factor receptor (EGFR) inhibiting therapy.
• Have experienced grade 3 or above toxicity from prior anti-PD1 therapy.
• Have p16 negative squamous cell carcinoma of unknown primary in cervical lymph node.
• Patients with primary nasopharynx or salivary gland cancers.
• Patients who have had chemotherapy, biological therapy or definitive radiation within 4 weeks of the study enrollment or those who have not recovered from adverse events to ≤ Grade 1 due to agents administered more than 4 weeks earlier.
• Had undergone any major surgery within 4 weeks of study enrollment.
• Had undergone any palliative radiation within 2 weeks of study enrollment.
• Have had other investigational agents within 4 weeks or 5 half-lives, whichever is shorter, of the study enrollment.
• Have known leptomeningeal metastases or untreated or symptomatic brain metastases. Treated, asymptomatic brain metastasis can be included.
• Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, autoimmune disease requiring systemic steroids, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements.
• Have a condition requiring systemic treatment with either corticosteroids (> 10 mg daily prednisone equivalents) or other immunosuppressive medications within 14 days of study drug administration. Inhaled or topical steroids and adrenal replacement doses > 10 mg daily prednisone equivalents are permitted in the absence of active autoimmune disease.
• Have clinically relevant coronary artery disease or history of myocardial infarction in the last 12 months or high risk of uncontrolled arrhythmia or uncontrolled cardiac insufficiency.
• Have uncontrolled or poorly controlled hypertension (>180 mmHg systolic or > 130 mmHg diastolic) at the time of enrollment.
• Prior treatment with a combination of cetuximab and a PD-1/PD-L1 inhibitor. Prior treatment with cetuximab or a PD-1/PD-L1 inhibitor is allowed as long as not previously given in combination.
• A history of allergic reactions attributed to compounds of chemical or biologic composition similar to those of cetuximab and/or nivolumab.
• Pregnant or breast-feeding.
• Known active HIV, Hep B, or Hep C infection. If not clinically indicated, the patients do not need to be tested.