Civil servants swear an oath to defend the constitution, not the political interests of whoever happens to be in office currently. Anyone who doesn't want to live in a banana republic should strongly support that.
I doubt there's a legal basis for saying that these "alt_X" accounts are functioning in the role of a federal worker. They're obviously not advocating the official government position -- that's the whole point of them. They clearly state they aren't reflective of the views of the agency they're referencing. You'd effectively be arguing that no government employee could ever, in any capacity, express an opinion related to the function of their agency.
This is a ridiculous statement. Who isn't a third party in an investigation, other than the investigated and the investigators? Bank records, third party statements, etc. This is one of those statements to save for posterity.
Everyone, gov. employees included, have first amendment rights. However they do not have the right to have their own agenda in their role as a government worker. That flies in the face of the core principle of the constitutional election of legislative and executive officers.
(If you want an example of twitter being inconsistent, just compare the suspension of Glenn Reynolds - instapundit - for tweeting "Run. Them. Down." in response to rioters on I277 and the lack of action on the multiple and explicit death threats the are tweeted regularly.)