You have rights - as long as you don't use them.

Imagine a political activist (Mr. X) who calls a politician (Mr. Z) a traitor. Furious, the government passes a law making it illegal to call public officials traitors.

Instead of getting angry at the government, the public turns on Mr. X: "If you had just kept your mouth shut, we'd still have that right!"

What they're effectively saying is we have certain rights in theory, which is really cool - just as long as we don't actually use those rights.

Does the public's argument constitute a fallacy? If so, what kind?

Wouldn't this be a false cause? He's done nothing wrong, and at worst is a catalyst.