You could answer them without completely spilling your guts. I often vomit from stress and anxiety. When it happens and people notice I simply say, "I have a weak stomach" or something equally vague instead of going on about my issues with anxiety. Perhaps if you keep it simple and say something along the lines of, "I don't do well with public speaking" or "I have a medical concern of a personal nature." Give them as much information as you feel comfortable with, but don't feel obligated to give more.

I hope it goes well for you