Roosevelt openly opposed bargaining rights for government unions.

"The process of collective bargaining, as usually understood, cannot be transplanted into the public service," Roosevelt wrote in 1937 to the National Federation of Federal Employees. Yes, public workers may demand fair treatment, wrote Roosevelt. But, he wrote, "I want to emphasize my conviction that militant tactics have no place" in the public sector. "A strike of public employees manifests nothing less than an intent on their part to prevent or obstruct the operations of Government."

RealClearPolitics - FDR's Ghost Is Smiling on Wisconsin's Governor

Why not have a union as a member of the military? Same reason as stated above.

Private unions create profits and have competition. Public unions are monopolies that raise taxes.