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 =\< institutional upper limit of normal (ULN)
• •Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 3 x institutional ULN
• •Creatinine =\< institutional ULN OR glomerular filtration rate (GFR) \>= 60 mL/min/1.73 m\^2 unless data exists supporting safe use at lower kidney function values, no lower than 30 mL/min/1.73 m\^2
• •Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial
• •Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, patients should be class 2B or better
• •Patients must be willing to comply with birth control requirements: The effects of the agents in this study (or similar agents) on the developing human fetus are either unknown, or known to be teratogenic, embryotoxic, and fetotoxic in mice and rabbits. For this reason, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and for 6 months after completing treatment administration. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. Men treated or enrolled on this protocol must also agree to avoid donating sperm for during the study period, and to use adequate contraception prior to the study, for the duration of study participation, and for 6 months after completion completing treatment administration
• •Patients must have the ability to understand and willingness to sign a written informed consent document. Participants with impaired decision-making capacity who have a legally-authorized representative and/or family member available will also be eligible