Union Integrity Act This bill establishes whistleblower protections for union employees. It prohibits a labor organization from discriminating against any of its employees who (1) provide information to the labor organization, the Department of Labor, or any other state, local, or federal government authority or law enforcement agency regarding any violation of the Labor-Management Reporting and Disclosure Act of 1959 or any Labor, National Labor Relations Board (NLRB), or National Mediation Board (NMB) order; (2) testify in any Labor, NLRB, or NMB administrative or enforcement proceeding; (3) file or institute any such proceeding; or (4) refuse to perform an assigned task that the employee reasonably believes is a violation of any law, order, or prohibition enforceable by Labor, the NLRB, or the NMB.