Has the US Supreme Court ever ruled on the right of government unions to exist.
I think the 10th Amendment answers that. If a power is not delegated to fedgov and is not otherwise repugnant to the Constitution, the states pretty much have a free hand.
They must have, because according to the links in the main article, FDR actually said... on August 16, 1937
"Particularly, I want to emphasize my conviction that militant tactics have no place in the functions of any organization of Government employees. Upon employees in the Federal service rests the obligation to serve the whole people, whose interests and welfare require orderliness and continuity in the conduct of Government activities. This obligation is paramount. Since their own services have to do with the functioning of the Government, a strike of public employees manifests nothing less than an intent on their part to prevent or obstruct the operations of Government until their demands are satisfied. Such action, looking toward the paralysis of Government by those who have sworn to support it, is unthinkable and intolerable. It is, therefore, with a feeling of gratification that I have noted in the constitution of the National Federation of Federal Employees the provision that "under no circumstances shall this Federation engage in or support strikes against the United States Government."
Clearly, some time after 1937, there was a watershed change, either in the perception of political leadership, or in the laws prohibiting public employee unions.