Medicare Buy-In and Health Care Stabilization Act of 2021 This bill establishes a Medicare buy-in option for certain qualifying individuals and makes a series of other changes relating to health care costs. Specifically, the bill allows individuals aged 50 to 64 to enroll in Medicare if such individuals would otherwise qualify for Medicare at the age of 65. The Centers for Medicare & Medicaid Services (CMS) must determine enrollment periods and set premiums for the buy-in option established under the bill, in accordance with specified requirements. The CMS must also award grants to states and nonprofit organizations for outreach and enrollment activities relating to the buy-in option. The bill also (1) establishes a supplemental option under Medicare to cover cost-sharing for beneficiaries; (2) repeals provisions that prohibit the CMS from negotiating the prices of prescription drugs; (3) establishes an individual market reinsurance program relating to coverage of high-cost individuals, as specified; and (4) makes permanent certain increases and expansions of the premium tax credit.