It's been awhile since I've played and I can remember reading something about not being able to regenerate while based by an opposing character. I reread the rules recently and I can't find anything that says that I can't regenerate while based by opposing characters. Basically, my question is, can I regenerate while based by opposing characters or do I have to break away?

You can't use support on someone who is adjacent to an opposing character. There's nothing to stop trying to Regen, except turning the power off, of course.