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

Inclusion Criteria: * Have an Eastern Cooperative Oncology Group (ECOG) performance status of =\< 2 at the time of study treatment initiation * Have pathologically confirmed diagnosis of NSCLC (Phase I, Phase II Studies A, C and D, E and Expansion Cohort) or squamous cell head and neck cancer (Phase II Study B) * Must be eligible for treatment with nivolumab as standard of care (for nivolumab treatment groups only) * Phase II Study A and Expansion Cohort AE: Patients with advanced (metastatic) NSCLC, whose disease progressed during or after platinum based therapy * Phase II Study B: Patients with advanced recurrent head and neck squamous cell carcinoma * Phase II Study C: Patients with unresectable NSCLC with PD-L1 expression \>= 50% for first line therapy in advanced stage. In the rare event that there is a discrepancy in the results of PD-L1 testing (i.e. 2 or more specimens were tested, etc.), eligibility status will be per the discretion of the principal investigator (PI) after review of other available biomarker testing * Phase II Study D: Patients with advanced squamous NSCLC with PD-L1 expression \<50% with PR/CR or stable disease by RECIST after at least 4 cycles of induction chemoimmunotherapy with platinum-based combination with pembrolizumab, prior to initiation of maintenance pembrolizumab * Phase II Study E: Patients with advanced NON-Squamous NSCLC (without EGFR/ALK/ROS-1/KRAS mutations) with PD-L1 expression \<50% with PR/CR or stable disease by RECIST after a…