Medicare Crisis Program Act of 2020 This bill makes a series of changes to health insurance coverage and requirements during the public health emergency relating to COVID-19 (i.e., coronavirus disease 2019). For example, the bill makes several changes to Medicare and Medicare Advantage (MA) during the public health emergency. Among other things, the bill allows all individuals who receive unemployment benefits, and their dependents, to enroll in Medicare if they otherwise do not have health insurance coverage; eliminates cost-sharing for COVID-19 drugs and treatment services under Medicare and MA; eliminates premiums under Medicare and caps overall cost-sharing under Medicare and MA; and establishes a national clearinghouse for the purchase and distribution of COVID-19 personal protective equipment for Medicare providers. The bill also alters requirements under Medicaid and the Children's Health Insurance Program (CHIP) and for private health insurers during the emergency. Among other things, the bill requires coverage of COVID-19 vaccines and treatment services under Medicaid and CHIP without cost-sharing; increases the Medicaid Federal Medical Assistance Percentage (FMAP) that is otherwise applicable during the public health emergency, including for states experiencing economic downturns (i.e., high unemployment); requires private health insurers to cover COVID-19 treatment services without cost-sharing; and requires federal payment to health care providers for the cost of COVID-19 treatment services that are provided to uninsured individuals, based on Medicare payment rates.