I can't even understand how a@@es like that can be supervisory positions. I'm sure there's a whole slew of reasons to report her to HR.
But after all, we all need our jobs.
Your lifestyle isn't her business. She's your boss for work, but the boss of you.
There's no reason to explain anything. If you know the place you'll be eating, check the menu ahead, even call if you have questions about ingrediants. I wouldn't ask about things like the soup having meat broth in front of her.
If she comments on the lack of meat, just answer what you need to.
Mostly it going to be related on how well you get along with her anyway. I can't imagine a food choice affecting that, unless it's complete departure from your norm.