How would the FA actually know if an illness was contagious? Also, another person's medical history is none of your business. It's private information.
If I heard an FA discuss my medical condition (not that I currently have one, but) they would be getting spoken to by the company very seriously.
Also. since this seems like a genuine reason to be moved by the FA why not seek to get a better seat/s?
If you have to ask the FA to be moved it should stand to reason they will ask why and I would think any FA out of concern for other passengers has the right to ask if it is contagious as do any passengers they may be moved next to but that doesn't mean divulging medical history or anything.