Can they cook my steak medium rare or is it now illegal in Utah?

Yes, rare even if you want

I would think they should be able to cook it any way you want. I understand that probably most people do medium well around here. So, what I would do is make sure you bring it to your server's attention that you really need it to be medium rare. But, another reason why some places don't like to do rare stuff is because sometimes it's better to cook the meat longer to make sure there's no bacteria problems.