You don't let your government "unionize" against you, I wouldn't want the Army to choose when it wants to go on strike against the people and call a work stoppage, and I sure don't want them joining some "international" labor movement against us.
I don't either. But neither do I wish to limit who another American associates with. If a soldier wishes to join a union, I could care less. However, I don't want the US government entering into a contract with a "soldier's union."
In the same way, I could care less if a public employee joins a union... he has the freedom of association. However, I do not want the US government to require union membership as a condition for employment.