All of which is to say that universities, even public universities, are not government entities in the same way that public schools are. It is far from clear to me that libertarians would support the idea of a private citizen working for a non-government organisation being compelled to hand over private communications by the government.
Of course I might be wrong about UVAs status, but I did a fair amount of searching, and could not find anything that would indicate that university employees were in fact government employees, and the information that I did find tended to support the idea that they were not. If I'm wrong about this, then the OP's position is easier to understand...