I'm not sure how to respond to this issue as I don't believe that I have a mental illness (I might have, but just don't realise it) but I do have physical disabilities that some who suffer the same may feel embarrassed by their inabilities to do what healthy people can do without even thinking about it.
If we are struck down by something, then we have to try and create a balanced atmosphere around ourselves that ensures that we understand our limitations and frustrations, but just as important that those around us and who have to live and work with us also understand our limitations and frustrations and how we can often lash out through sheer frustration. It's never really anything personal against any individual or group of people, just that we have ailments that require understanding and not revulsion.
There have been times when I used to feel embarrassed because I couldn't get into a taxi because it was too high, or not being able to manoeuvre around an obstacle - now I tend to make a joke of it to lighten the mood. I'm not sure if such an attitude could be applied to someone with mental illness. I believe that people need to assess what is meant by mental illness before being judgemental. There are many forms of mental illness and I suppose it can be scary for some people who have to deal with someone else's afflictions when they don't understand what is wrong.
Support and understanding go a long way and if more people took the time to realise this, then perhaps embarrassing situations could be avoided.