If the bathroom is empty, then I would still take my son in with me. If there were other women and/or girls in the bathroom then I would let him go into the men's room and I wait outside the door. This started when he was five or six. My son was actually the one that wanted to go in the men's room at that age.
I did stand guard outside the door, and occasionally, a guy (most likely a dad) would come out and ask if that was my son in there and let me know that he was doing fine and would be out in a minute.
Thankfully, I've never been in a position where I would have to go in after him, but I wouldn't hesitate to go in after him if I felt something was wrong. It doesn't hurt to have a conversation with him about what is okay and what isn't when he's by himself in the bathroom.
Sounds like you have an independent little man on your hands! That's awesome! Hope this helps. Good luck.