"Stay well" means, "don't get sick". "Stay safe" means, "don't get into a dangerous situation.
With the coronavirus panic going around, people are saying "Stay well" and/or "Stay safe" to mean, "I hope you don't catch this disease" or "I encourage you to take the necessary precautions to avoid catching this disease". Before this, I occasionally heard people say "Stay safe" when people were leaving to engage in a potentially dangerous activity. Like when the coach finishes his lecture to the team and tells them to go out and practice a rough sport, he would sometimes say "stay safe". But it was pretty rare before this coronavirus stuff, at least in my personal experience.
Sure, you could say "be well" or "be safe", and it would mean essentially the same thing. If someone is already sick, then "stay well" wouldn't make much sense because they can't continue to do something they're not doing now. But saying "be well" to someone who is sick would seem a little lame. People sometimes say, "I hope you get well" or the like, which I think is weak also, but whatever. Anyway, people rarely say "be well" or "be safe" as a good-bye. So you could say it if you meant it literally, but it would be unusual.