Testing the Addition of an Anti-cancer Drug, Berzosertib (M6620, VX-970), to the Usual Treatments (Carboplatin and Gemcitabine) and to Pembrolizumab for Patients With Advanced Squamous Cell Non-small Cell Lung Cancer

Inclusion Criteria: * Patients must have histologically confirmed NSCLC of predominantly squamous cell histology, stage IV (American Joint Committee on Cancer \[AJCC\] 8th edition) * Patients must have measurable disease, as defined by Response Evaluation Criteria in Solid Tumors (RECIST) version (v)1.1 * Patients must have tumor tissue available at time of enrollment, or be willing to undergo a biopsy for integrated biomarker studies * Patients with a history of prior platinum-based systemic chemotherapy given as neoadjuvant, adjuvant, or consolidation therapy for early stage or loco-regionally advanced NSCLC are eligible, if therapy is completed one year prior to initiation of treatment. Patients must not have had prior systemic chemotherapy or immunotherapy for metastatic disease * Patients with prior immunotherapy given as neoadjuvant, adjuvant, or consolidation therapy for early stage or loco-regionally advanced disease are eligible, if treatment is completed one year prior to initiation of treatment * Patients must have Eastern Cooperative Oncology Group (ECOG) performance status of 0-1 (Karnofsky \> 60%) * Age \>= 18 years. Because no dosing or adverse event data are currently available on the use of these drug combinations in patients \< 18 years of age, children are excluded from this study, but will be eligible for future pediatric trials * Leukocytes \>= 3,000/mcL * Absolute neutrophil count \> lower limit of normal (LLN) * Platelets \> LLN * Total bilirubin =\< i…