American Jobs First Act of 2021 This bill imposes additional requirements related to the H-1B (specialty profession) nonimmigrant visas and repeals various immigration-related programs. The bill repeals the diversity visa program, which makes immigrant visas available to aliens from countries with historically low rates of immigration to the United States. The bill also eliminates a program that provides temporary employment authorization to an F-1 student visa holder during or after the completion of the student's studies, unless Congress expressly authorizes such a program. The bill's provisions relating to the H-1B program include requiring an H-1B employee to receive a wage that is at least the wage of the U.S. citizen or permanent resident who had the same job in the two years before the H-1B application was filed, where currently an H-1B worker may be paid the prevailing wage for the occupational classification in the area of employment; prohibiting an employer from hiring an H-1B employee if the employer has involuntarily separated without cause an employee in a substantially similar occupation in the two years before filing the H-1B application or will do so in the two years after the filing; prohibiting an employer from hiring an H-1B employee if there had been a strike or lockout related to salary or benefits in the two years before filing the H-1B application; increasing the maximum penalties for violating certain H-1B requirements; and prohibiting in some instances the use of nondisclosure agreements to prevent an employee from disclosing an employer's possible misuse of the H-1B program.