Perhaps instead of going on the offense right away, you should ask your coworkers how it is that they came to have nicknames on their nametags. Different places have different style nametags, so it's not always as simple to change it. I've had some that were engraved in hard plastic, and others that had an insert section for laminated paper. It would have been much easier to change that latter.
I'd personally see how they went about it, and then take that knowledge to your supervisor. If they push back after that, go straight to HR and explain the issue, because they've then crossed a line.
Do they collect the badges when people leave? Some of the places I've worked had a container of old ones. If you had forgotten yours and didn't want to put a sticker on, you could pick another name. Ben is pretty common if that's what you're looking for.
