STD and workers comp are mutually exclusive. Workers comp is for work related illnesses and injuries; STD is for non-work-related illnesses and injuries. If you are eligible for the one, it means you are not eligible for the other. You will not be able to use both for the same injury.
Two things I am tired of typing: 1.) A wrongful termination does not mean that you were fired for something you didn't do; it means that you were fired for a reason prohibited by law. 2.) The above answer, whatever it is, assumes that no legally binding contract or CBA expressly says otherwise. If it does, the terms of the contract apply.