The democrats aren't why the squirrel is dead, be serious. Animal controls regulations are the same for animal bites and rabies risk in pretty much every state and the call supposedly came from a lady from texas.

Most states, including red ones, have laws against owning a squirrel

You aren't wrong, but I think the point here is the more general notion of "you can't trust the government/the government isn't your friend/the government will screw you over". And there, on that more general concept, there is much more of a partisan divide, with dems generally saying "you don't need a gun because you have the government" and reps saying "guns should be legal because you can't fully trust the government". That's a generalization, of course, but the tendency is there.