To a lot of Trump supporters Clinton does have a mental illness in that she is a woman. As bad as she could be -- the only alternative that could win is a lot worse.
I have heard no evidence that she has a mental illness although we know that one in five people do and I would guess among Presidents it is likely higher than that. And we know that many highly effective world leaders have suffered from mental illness.
I suspect she is getting whatever health care support she needs as she can afford it.
As for Trump -- he is a very nasty, prejudiced man with a horrible outlook -- it does not matter if he is ill or not -- someone with those views should not lead any country. He may well be just a prejudiced turd who is not very bright, rather in love with himself, but otherwise healthy in a clinical sense. I would be more comfortable with someone fighting personal demons that someone who is fighting half the world due to his hatred.