This phase I trial studies the side effects and the best dose of WEE1 inhibitor MK-1775 when given together with docetaxel and cisplatin in treating patients with stage III-IVB squamous cell carcinoma of the head and neck that may or may not be able to be removed by surgery (borderline resectable). WEE1 inhibitor MK-1775 may block the growth of tumor cells by blocking some of enzymes that are needed for tumor growth and may also help docetaxel and cisplatin work better by making tumor cells more sensitive to the drug. Drugs used in chemotherapy, such as docetaxel and cisplatin, work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Giving WEE1 inhibitor MK-1775 with docetaxel and cisplatin before surgery may kill more tumor cells and shrink the tumor, allowing patients to undergo surgery to remove it.