Sounds like some punk ass kid just want to "street fight" and act tough. (Apology if you aren't)
If you are defending yourself/others, you won't have the freaking time to put a month guard on, especially if you have braces, which takes extra time to fiddle the mouth guard to fit in. If you have that amount of room and time to do so, go call the police/grab a weapon/run instead if you are alone.
To answer your question: Yes, there are multiple different style of mouth guards specially designed for braces that are good. Normal ones don't fit if you are wearing the traditional metal frame braces.
However if you have braces, the better advice is don't act tough and "fight". (Your teeth are "moving", so they are less stable than usual anyway.) If you really need to defend yourself/others, carry something actually useful than other than a mouthguard with you LOL. Especially if you have had no training and experience fighting with bare hand, a mouth guard is going to be one of the least useful thing for you. LOL
Even for trained fighters, no one will carry a mouth gaurd all the time in purpose to actually defend themselves, 'cause they'd be worrying about knifes, bottles and even guns instead, mouth guard isn't gonna be the useful item to carry for self-defense in any cases. Go grab a chair/use your jacket instead of putting a mouth guard on with that time. 'Cause you would be really fighting for your life, not your teeth only.
Carry stuff like self defence paint spray, pepper spray(if legel), taser(if legel), stun gun(if legal), baton, a flashlight or even an alarm or some other practical self defend equipment. And you can easily ask your female friends or whoever else isn't as physically strong to carry one as well, these item will still be as useful in their hands. More numbers you have on your side is always better. Not carrying a mouth guard, come on now...