CIMAvax Vaccine, Nivolumab, and Pembrolizumab in Treating Patients With Advanced Non-small Cell Lung Cancer or Squamous Head and Neck Cancer

PRIMARY OBJECTIVES: * I. To identify the maximum dose of CIMAvax in combination with nivolumab based on dose limiting toxicities (DLTs) as assessed by Common Terminology Criteria for Adverse Events version 4.03 (CTCAE version \[v\] 4.03). (Phase I) * II. To evaluate the 12-month overall survival of CIMAvax combined with nivolumab in patients with advanced non-small cell lung cancer (NSCLC). (Phase II-Study A) * III. To evaluate the 6-month progression free survival (PFS) of CIMAvax combined with nivolumab in patients with advanced recurrent squamous cell carcinoma of the head and neck. (Phase II-Study B) * IV. To evaluate the objective response rate of pembrolizumab in combination with CIMAvax as first-line therapy in patients with advanced NSCLC (PD-L1 expression \>= 50%). (Phase II-Study C) * V. To evaluate the 12-month PFS of pembrolizumab in combination with CIMAvax as maintenance therapy in patients with advanced squamous NSCLC (PD-L1 expression \< 50%) (Phase II-Study D) * VI. To evaluate the 12-month PFS of pembrolizumab in combination with CIMAvax as maintenance therapy in patients with advanced NON-squamous NSCLC without EGFR/ALK/ROS-1/KRAS mutations (PD-L1 expression \< 50%) (Phase II- Study E) SECONDARY OBJECTIVES * I. To assess the toxicity of CIMAvax combined with nivolumab using the Cancer Therapy Evaluation Program (CTEP) National Cancer Institute (NCI) Common Terminology Criteria for Adverse Events (CTCAE version 4.03). (Phase I) * II. Determine the preliminary efficacy of the combination of anti-PD1 therapy with CIMAvax. (Phase I) III. To evaluate progression free survival (PFS) for the combination of CIMAvax and nivolumab in patients with advanced NSCLC. (Phase II-Study A) IV. To evaluate the 12-month overall survival of patients with advanced recurrent squamous cell carcinoma of the head and neck who received nivolumab in combination with CIMAvax. (Phase II-Study B) * V. To evaluate the PFS and 12-month overall survival of CIMAvax in combination with pembrolizumab as first-line therapy in patients with advanced NSCLC (PD-L1 expression \>= 50%). (Phase II-Study C) * VI. To evaluate the PFS and overall survival of CIMAvax in combination with pembrolizumab as maintenance therapy in patients with advanced squamous NSCLC (PD-L1 expression \< 50%). (Phase iI- Study D) * VII. To evaluate the PFS and overall survival of CIMAvax in combination with pembrolizumab as maintenance treatment in non-squamous NSCLC patients without EGFR/ALK/ROS-1/KRAS mutations after induction chemoimmunotherapy (PD-L1 expression \< 50%).(Phase II - Study E) * VIII. To assess the toxicity of CIMAvax combined with nivlumab or pembrolizumab suing the CTEP NCI Common Terminology Criteria for Adverse Events (CTCAE version 4.03). (Phase II) TERTIARY OBJECTIVES: I. To conduct correlative studies comparing blood EGF levels, platelet levels, markers of immune response and functionality of antibody response. (Phase I) II. To examine the association of EGFR (total and activated), PD-1 and PD-L1 expression and mutations in tumor tissue with biomarkers of genetic and immune response. (Phase I and II) III. Comparison of response assessment criteria for a prospective analysis (immune-related \[ir\] Response Evaluation Criteria in Solid Tumors \[RECIST\] response assessment versus \[vs.\] immune-related Response Criteria \[irRC\] vs. RECIST 1.1). (Phase I and II) IV. To characterize the blood EGF levels and other blood-based biomarkers of patients censored from the trial because of low titer response. (Phase II) OUTLINE: This is a phase I dose escalation study of CIMAvax followed by a phase II study. LOADING PHASE I: Patients receive CIMAvax intramuscularly (IM) and nivolumab intravenously (IV) over 60 minutes on day 1. Treatment repeats every 2 weeks for up to 4 doses in the absence of disease progression or unacceptable toxicity. Within 4 weeks after the 4th dose, patients receive CIMAvax IM at the same time as the next nivolumab dose. MAINTENANCE PHASE I: Patients who do not experience a DLT receive CIMAvax every 4 weeks and nivolumab every 2 weeks. PHASE II STUDY A and B: Patients receive CIMAvax IM and nivolumab IV over 60 minutes. Treatment with CIMAvax repeats every 2 weeks for 4 doses during the loading phase and every 4 weeks during the maintenance phase in the absence of disease progression or unacceptable toxicity. Courses for nivolumab repeat every 2 weeks in the absence of disease progression or unacceptable toxicity. Patients in Study A with antibody titer \>= 1:4000 at the end of the loading phase may receive CIMAvax IM every 8 or 12 weeks during the maintenance phase. PHASE II STUDY C: Patients with PD-L1expression \>= 50% receive CIMAvax IM and pembrolizumab IV over 30 minutes. Treatment with CIMAvax repeats every 2 weeks for 4 doses during the loading phase and every 4 weeks during the maintenance phase in the absence of disease progression or unacceptable toxicity. Courses for pembrolizumab repeat every 2 weeks for 2 years in the absence of disease progression or unacceptable toxicity. PHASE II STUDY D: Patients with PD-L1 expression \< 50% after 4 cycles of induction chemotherapy with pembrolizumab, receive CIMAvax IM and pembrolizumab IV over 30 minutes. Treatment repeats every 4 weeks for 2 years in the absence of disease progression or unacceptable toxicity After completion of study treatment, patients are followed up every 30 days for 120 days.