National Signing Bonus Act of 2021 This bill replaces existing Federal Pandemic Unemployment Compensation payments for unemployed individuals with two-time signing bonuses for newly employed individuals. Specifically, the bill allows states to provide up to two direct payments to individuals who are eligible for Federal Pandemic Unemployment Compensation prior to commencing employment with a new employer. The individual must continue employment with the new employer for at least four weeks before July 4, 2021, to be eligible for the first $1,212 payment under the bill and for at least eight weeks before July 4, 2021, to be eligible for the final $1,212 payment. The bill otherwise eliminates federal reimbursement for Federal Pandemic Unemployment Compensation payments.