Building America's Health Care Workforce Act This bill extends certain flexibilities relating to training and competency requirements for nurse aides in Medicare skilled nursing facilities (SNFs) and Medicaid nursing facilities (NFs). Generally, SNFs and NFs may not hire nurse aides for longer than four months unless they meet certain training and certification requirements. The Centers for Medicare & Medicaid Services waived some of these requirements during the COVID-19 emergency period so as to allow facilities to maintain adequate staffing levels; the waiver ends in June 2022. The bill extends these flexibilities for at least two years after the bill's enactment and specifies that (1) any time worked by a nurse aide during the emergency period counts toward the 75 hours required for initial training, and (2) a competency evaluation may be conducted at the SNF or NF where the nurse aide is employed if the state does not offer such an evaluation at least once a week.