Employees of the federal government by law are guaranteed the right to refrain from assisting a union. See 5 U.S.C. § 7102 [1] (federal employees generally); 39 U.S.C. § 1209(c) [2] (postal employees). (U.S.C. stands for "United States Code." To access the United States Code, click here [3].)
This means that you cannot be required to pay any dues or fees to the union in violation of your religious beliefs.
Links:
[1] http://www4.law.cornell.edu/uscode/5/7102.shtml
[2] http://www4.law.cornell.edu/uscode/39/1209.shtml
[3] http://www4.law.cornell.edu/uscode/