Neodjuvant Nivolumab and Lirilumab, Followed by Surgery, Followed by Adjuvant Nivolumab and Lirilumab, in SCCHN

Inclusion Criteria: * Subject must have histologically or cytologically confirmed locoregionally recurrent squamous cell carcinoma of the head and neck (including any primary site, such as oral cavity, oropharynx, larynx or hypopharynx, and nasopharyngeal carcinoma) * Must be a candidate for salvage surgery * Willing to provide blood and tissue from diagnostic biopsy and at the time of surgery * Has documented disease-free interval (DFI) \> 8 weeks after completion of initial therapy; DFI is from the time of completion of initial treatment to the diagnosis of local or locoregional recurrence * Any HPV status or smoking history is permitted. Oropharyngeal cancer patients are required to undergo HPV testing with p16 immunohistochemistry and/or confirmatory HPV PCR or ISH testing * Age 18 years or older * ECOG performance status ≤2 (Karnofsky ≥60%, see Appendix A) * Participant must have normal organ and marrow function as defined below within 21 days prior to study registration: * leukocytes ≥3,000/mcL * absolute neutrophil count ≥1,500/mcL * platelets ≥100,000/mcL * total bilirubin ≤2.0 g/dL * AST(SGOT)/ALT(SGPT) ≤2.5 × institutional upper limit of normal * creatinine within normal institutional limits OR * creatinine clearance ≥60 mL/min/1.73 m2 for participants with creatinine levels above institutional normal * Ability to understand and the willingness to sign a written informed consent document * Women of childbearing potential (WOCBP) must agree to use appropriate method(s) of contraception. WOCBP should plan to use an adequate method to avoid pregnancy for 5 months (30 days plus the time required for nivolumab to undergo five half-lives) after the last dose of investigational drug * Women of childbearing potential must have a negative serum or urine pregnancy test (minimum sensitivity 25 iu/l or equivalent units of hcg) within 24 hours prior to the start of nivolumab * "Women of childbearing potential (WOCBP)" is defined as any female who has experienced menarche and who has not undergone surgical sterilization (hysterectomy or bilateral oophorectomy) or who is not postmenopausal. Menopause is defined clinically as 12 months of amenorrhea in a woman over 45 in the absence of other biological or physiological causes. In addition, women under the age of 55 must have a documented serum follicle stimulating hormone (FSH) level less than 40 mIU/mL * Men who are sexually active with WOCBP must agree to use any contraceptive method with a failure rate of less than 1% per year. Men who are sexually active with WOCBP will be instructed to adhere to contraception for a period of 7 months after the last dose of investigational product. Women who are not of childbearing potential (ie, who are postmenopausal or surgically sterile as well as azoospermic men) do not require contraception Exclusion Criteria: * Existing significant autoimmune conditions. Patients with a history of Hashimoto thyroiditis who are stable on replacement hormone therapy are not excluded. Patients cannot be on long-term (\> 4 weeks) corticosteroids at doses exceeding prednisone 20 mg (or its equivalent) prior to enrollment. Short-term corticosteroid dosing is permitted as long as steroids are discontinued within 2 weeks of study enrollment. * Subject who has had chemotherapy or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who have not recovered from adverse events due to agents administered more than 4 weeks earlier. * Subject who has been treated with immunotherapy. This includes prior treatment with anti-PD-1, anti-PD-L1, anti-PD-L2, anti-CTLA-4 antibody, or any other antibody or drug specifically targeting T-cell co-stimulation or checkpoint pathways * Subject with known brain metastases should be excluded from this clinical trial because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events. However, baseline brain imaging is not required prior to enrollment in the study if patients are asymptomatic. * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements. * Known human immunodeficiency virus carrier or a diagnosis of immunodeficiency. Any positive test result for hepatitis B virus or hepatitis C virus indicating presence of virus, eg, Hepatitis B surface antigen (HBsAg, Australia antigen) positive, or Hepatitis C antibody (anti-HCV) positive (except if HCV-RNA negative). * Known non-infectious pneumonitis or any history of interstitial lung disease. * Receipt of a live vaccine within 30 days of start of study treatment