If someone wants to prove that they voted, great they have their stubs. Take a picture of that. If someone can prove how they voted, then some will be forced to do so, by abusive spouses or employers. Failure to vote the right way, or even to take your selfie, will grounds for a beating or dismissal.
New Hampshire isn't the only state to have that law. I believe it's the law in every state. TechDirt needs to do some more reading on elections before it spreads disinformation like this.