I think the best thing you can do is continue to be supportive.

It could be that with her "great" upbringing she didn't know much about these illnesses or heard that they were bad or weaknesses rather than actual medical issues...a lot of people grew up hearing that if it wasn't prevalent in their lives.

You can't make somebody accept something they aren't ready to accept, but you can be there for her to help her understand and to deal with it.

I hope that helps.