Inclusion Criteria:

* Patients must meet all SCREENING/PRE-SCREENING and SUB-STUDY REGISTRATION COMMON ELIGIBILITY CRITERIA as specified in S1400: Phase II/III Biomarker-Driven Master Protocol for Previously Treated Squamous Cell Lung Cancer (Lung-Map)
* Patients must be assigned to S1400C
* Patients must not be taking within 7 days prior to sub-study registration, nor plan to take while on protocol treatment and for 14 days after the last dose of study treatment, strong CYP3A4 inhibitors and/or strong CYP3A4 inducers; moderate inhibitors or inducers of isoenzyme CYP3A4 should be avoided, but if necessary can be used with caution
* Patients must not be taking within 7 days prior to sub-study registration, nor plan to take while on protocol treatment drugs that are known to prolong the QT interval
* Patients must not have a screening corrected QT Fridericia?s formula (QTcF) interval \> 480 msec based on the average of the triplicate electrocardiograms (EKGs) performed within 28 days prior to registration; NOTE: triplicate EKGs are required at other timepoints; patients must not have any family or personal history of long or short QT syndrome, Brugada syndrome or known history of QTc prolongation, or torsade de pointes
* Patients must be able to take oral medications; patient may not have any impairment of gastrointestinal function or gastrointestinal disease that may significantly alter the absorption of palbociclib (e.g. ulcerative disease, uncontrolled nausea, vomiting, diarrhea, malabsorption syndrome or small bowel resection)
* Patients must have a sodium (Na), potassium (K), chlorine (Cl), calcium (Ca), magnesium (Mg), and glycosylated hemoglobin measurement (HbA1c) performed within 7 days prior to sub-study registration
* Patients must also be offered participation in banking for future use of specimens
* STEP 2 PALBOCICLIB RE-REGISTRATION:
* Patients must have progressed on Arm 2 (docetaxel) of this sub-study
* Patients must not have received any prior systemic therapy (systemic chemotherapy, immunotherapy or investigational drug) within 21 days prior to step 2 re-registration; patients must have recovered (\< grade 1) from any side effects of prior therapy
* Patients must have measurable disease documented by computed tomography (CT) or magnetic resonance imaging (MRI); the CT from a combined positron emission tomography (PET)/CT may be used to document only non-measurable disease unless it is of diagnostic quality; measurable disease must be assessed within 28 days prior to step 2 re-registration; pleural effusions, ascites and laboratory parameters are not acceptable as the only evidence of disease; non-measurable disease must be assessed within 42 days prior to step 2 re-registration; all disease must be assessed and documented on the Baseline Tumor Assessment Form; patients whose only measurable disease is within a previous radiation therapy port must demonstrate clearly progressive disease (in the opinion of the treating investigator) prior to registration
* Patients must have a CT or MRI scan of the brain to evaluate for central nervous system (CNS) disease within 42 days prior to step 2 re-registration; patient must not have leptomeningeal disease, spinal cord compression or brain metastases unless: (1) metastases have been locally treated and have remained clinically controlled and asymptomatic for at least 14 days following treatment and prior to re-registration, AND (2) patient has no residual neurological dysfunction and has been off corticosteroids for at least 24 hours prior to re-registration
* Patients must not be planning to receive any concurrent chemotherapy, immunotherapy, biologic or hormonal therapy for cancer treatment; concurrent use of hormones for non-cancer-related conditions (e.g., insulin for diabetes and hormone replacement therapy) is acceptable
* Patients must not have a screening QTcF interval \> 480 msec based on the average of the triplicate EKGs performed within 28 days prior to step 2 re-registration; NOTE: triplicate EKGs are required at other timepoints; patients must not have any family or personal history of long or short QT syndrome, Brugada syndrome or known history of QTc prolongation, or torsade de pointes
* Absolute neutrophil count (ANC) \>= 1,500/mcl obtained within 28 days prior to step 2 re-registration
* Platelet count \>= 100,000 mcl obtained within 28 days prior to step 2 re-registration
* Hemoglobin \>= 9 g/dL obtained within 28 days prior to step 2 re-registration
* Serum bilirubin =\< institutional upper limit of normal (IULN); for patients with liver metastases, bilirubin must be =\< 5 x IULN within 28 days prior to step 2 re-registration
* Either alanine aminotransferase (ALT) or aspartate aminotransferase (AST) =\< 2 x IULN within 28 days prior to step 2 re-registration (if both ALT and AST are done, both must be \< 2 IULN); for patients with liver metastases, either ALT or AST must be =\< 5 x IULN (if both ALT and AST are done, both must be =\< 5 x IULN)
* Patients must have a serum creatinine =\< the IULN OR measured or calculated creatinine clearance \>= 50 mL/min
* Patients must have a Na, K, Cl, Ca, Mg, and HbA1c performed within 7 days prior to sub-study registration
* Patients must have Zubrod performance status of 0-1 documented within 28 days prior to step 2 re-registration
* Patients must not have any grade III/IV cardiac disease as defined by the New York Heart Association Criteria (i.e., patients with cardiac disease resulting in marked limitation of physical activity or resulting in inability to carry on any physical activity without discomfort), unstable angina pectoris, and myocardial infarction within 6 months, or serious uncontrolled cardiac arrhythmia
* Patients must not have documented evidence of acute hepatitis or have an active or uncontrolled infection
* Patients with a known history of human immunodeficiency virus (HIV) seropositivity:

* Must have undetectable viral load using standard HIV assays in clinical practice
* Must have cluster of differentiation (CD)4 count \>= 400/mcL
* Must not require prophylaxis for any opportunistic infections (i.e., fungal, mycobacterium avium complex \[mAC\], or pneumocystis pneumonia \[PCP\] prophylaxis)
* Must not be newly diagnosed within 12 months prior to re-registration
* Pre-study history and physical exam must be obtained within 28 days prior to re-registration
* No other prior malignancy is allowed except for the following: adequately treated basal cell or squamous cell skin cancer, in situ cervical cancer, adequately treated stage I or II cancer from which the patient is currently in complete remission, or any other cancer from which the patient has been disease free for five years
* Patients must not be pregnant or nursing; women/men of reproductive potential must have agreed to use an effective contraceptive method; a woman is considered to be of "reproductive potential" if she has had menses at any time in the preceding 12 consecutive months; in addition to routine contraceptive methods, "effective contraception" also includes heterosexual celibacy and surgery intended to prevent pregnancy (or with a side-effect of pregnancy prevention) defined as a hysterectomy, bilateral oophorectomy or bilateral tubal ligation; however, if at any point a previously celibate patient chooses to become heterosexually active during the time period for use of contraceptive measures outlined in the protocol, he/she is responsible for beginning contraceptive measures
* As a part of the Oncology Patient Enrollment Network (OPEN) registration process the treating institution's identity is provided in order to ensure that the current (within 365 days) date of institutional review board approval for this study has been entered in the system
* Patients with impaired decision-making capacity are eligible as long as their neurological or psychological condition does not preclude their safe participation in the study (e.g., tracking pill consumption and reporting adverse events to the investigator)
* Patients must be informed of the investigational nature of this study and must sign and give written informed consent in accordance with institutional and federal guidelines