Health Care Improvement Act of 2021 This bill makes various changes to the health care marketplace, such as expanding premium assistance, encouraging states to expand Medicaid, and establishing a public health insurance option. First, the bill expands the premium tax credit that provides assistance, based on household income, for the cost of insurance plans purchased through an exchange. The bill also provides funding for state reinsurance payments to health insurance issuers and funding for insurance-related costs for individuals enrolled in plans through individual marketplaces. It also rescinds certain rules and guidance related to short-term, limited duration plans and the state innovation waiver program. Additionally, the bill begins the seven-year period of federal assistance for states expanding coverage under Medicaid when a state expands coverage and the bill reduces assistance to states that do not expand coverage. Further, the bill establishes a public health insurance option that is available through individual marketplaces. Providers participating in Medicare or Medicaid also must participate in the public option. The bill also makes changes with respect to payments for rural health providers and permits small group market plans to be offered across state lines. The bill authorizes the Department of Health and Human Services to negotiate with manufacturers for the prices of prescription drugs offered through Medicare. Finally, the bill establishes guidelines for employers to provide information about available health coverage and requires the Government Accountability Office to evaluate the employer notification process for the advance payment of premium tax credits.