I'm not some great logician or anything, but in its most basic framing "You don't need to worry about surveillance if you have nothing to hide" would be along the lines of a proving too much fallacy as the conclusion is much too broad for the argument of just having nothing to hide. As with a lot of informal fallacies (fallacies made due to content and/or context of the argument), you could probably ascribe a few of them to this statement, for example you could probably correctly state that this is a thought-terminating cliché as well.

Depending on how it is deployed, as described in one of the comments of the linked post, this could also constitute a formal fallacy (reasoning with a flaw in its structure), specifically denying the antecedent. As a TL;DR, the structure would have to be "If you have something to hide then you should worry about surveillance [if p then q], therefore if you have nothing to hide then you shouldn't worry about surveillance [if !p then !q]".

In my personal view call it a fallacy or not, the strongest arguments against "nothing to hide" have nothing to do with its fallacious nature or lack thereof. Additionally, demonstrating that an argument is fallacious just demonstrates that the argument needs to be reconstructed, rephrased, or better supported, not that its conclusion is false (else you fall victim to argument from fallacy, aka the fallacy fallacy).