Union Integrity Act This bill amends the Labor-Management Reporting and Disclosure Act of 1959 to prescribe whistle-blower protections for union employees. It prohibits a labor organization from discriminating against any of its employees who have: (1) provided 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 Act or any Labor or National Labor Relations Board (NLRB) order; (2) testified in any Labor or NLRB administrative or enforcement proceeding; (3) filed or instituted any such proceeding; or (4) refused to perform an assigned task that the employee reasonably believed to be in violation of any law, order, or prohibition enforced by Labor or the NLRB.