Do baseball umpires wear hats under the hocky-style masks?

Do NOT wear a backward hat while you umpire. Nothing displays the fact that you're a new amateur umpire faster and more clearly than a backward hat under your mask.

I really can't imagine trying to wear a hat under a hockey style mask/helmet. In fact I would say that if you can get the helmet on over a hat then the helmet is too big and is not providing adequate protection.

Remember that you need to be able to quickly remove the helmet using only one hand. If you can't do that with a hat, then don't wear a hat.

The reason you're getting different answers from your fellow umpires and supervisors is that there isn't a consensus. Some wear a cap, some don't, and some find that it doesn't work to try. Some think it doesn't look umpire-like to not have a cap.

If you want to try, it's probably easier with a four-seam cap. And a backwards cap in a hockey-style helmet isn't going to work.

BTW, there aren't many umpires here on Y!A. You probably already have better information from your colleagues than you will get from here.