You need to be very, very careful about how you do this, and I'm not sure I'd mention it in response to the particular secondary prompt you mentioned.
I wrote about recovering from an eating disorder in my PS (I didn't mention it in my secondaries though), and I think it was the right decision for me, personally. I got plenty of interviews, and while it's entirely possible that some of the schools that rejected me did so on the basis of my PS, I didn't encounter any blatant stigma.
I think the ONLY reason you should mention a mental illness is if it has shaped your interest in medicine somehow. This is why I would NOT mention it for this secondary prompt -- I don't think you should invoke a mental health issue in order to mitigate some part of your application that is sub-par. This information is very private, and you should only share it if it is really and truly relevant to your application.
If you do decide to mention your mental health issues, you need to make it very, very clear that the issue is resolved. Adcoms are going to be hesitant to take a chance on an applicant whose health may negatively influence their ability to treat patients. Some mental health disorders are perceived (rightly or wrongly) as being fairly short-term, or fairly chronic/likely to recur, and you need to make a judgement call as to whether your particular problem is going to be regarded as resolved or not. If you're dealing with something like schizophrenia, DO NOT EVEN THINK about mentioning it.
This is an issue that can be addressed, but you need to do so very carefully. I would suggest having your premed advisor read over a draft of whatever you choose to write -- he or she can give you (somewhat) informed advice. Again, for this particular secondary essay, I would NOT mention any mental health issues, but that's just my opinion.