This is a subject near to my heart. I knew quite a few JWs who committed suicide, my own daughter attempted twice.
I agree with Steve that it's impossible to know for a fact if the JWs have a higher suicide rate, but I think I can say from experience that their teachings and practices are often not helpful to those going through mental health issues. I can't blame the religion for my daughter's issues entirely, she probably would have had some issues no matter what religion she belonged to. But I also think it did contribute to her problems. I feel we were lucky to have made it through those years alive.
The religion engages in black and white thinking, they use guilt to manipulate people and their shunning practices are harsh. The elders might mean well, but seeing everything as a spiritual issue is not helpful to people who are suffering from depression. They are trained to determine repentance when dealing with issues of sin, when someone having mental issues might not express repentance in the way they expect, or might be too depressed to say what the elders want to hear. Some people take things to heart more than others, being disfellowshipped might be the end of the world to them, especially if they lose their entire support system because of it.
Someone who is a minister or counselor in a church would be expected to have a four year degree and advanced training or go to seminary, because it's important to know how to talk to those undergoing problems, the last thing you want to do is make The situation worse. The Watchtower is more interested in determining guilt and getting rid of those they deem unrepentant, they don't really seem can concerned about the impact this has on the individual.