I've heard that the Red Cross symbol is off limits...is this true?
Yes. It's literally illegal under 18 USC 706. It's not a trademark issue or anything like that, it came out of the Geneva Convention. All signatory nations agreed to protect that insignia under their laws. Of course, the law doesn't really protect something as specific as a boardgame component. It really just prevents its usage as part of company or product logos/insignias. Most things I've seen, like first aid kits, have gone to a green cross.