I can still remember my grandmother saying, "when you point a finger at someone, remember that three fingers are pointing at you." How wise, indeed. A large percentage of atheists I interact with on a regular basis are very skilled at pointing out flaws and weaknesses in other people's reasoning. They are very good at demonstrating how and why religious people often engage in tribalism or groupthink. And they often fail at recognizing these same problems in their own groups and circles.
First I'd like to address the claim itself, and then offer a bit of a verbal spanking for that kind of baseless assertion from a man in his position.
The National Institute of Health indicates that people who suffer from depression are more likely to turn to religion for assistance, and also more likely to break out of depression because of religious beliefs. The NIH indicates that the correlation between mental health deficiencies and religiosity are inconclusive due to a lack of data. (Link to the NIH Study)
The Psychiatric Times, which is a professional journal for psychiatrists, indicates that as of 2000, more than 724 studies have been done on the correlation between religiosity and mental health disorders. The journal is quoted thusly:
"The evidence suggests that, on balance, religious involvement is generally conducive to better mental health." - Psychiatric Times, January 2010
The Psychiatric Bulletin indicates that religious beliefs and practices may help patients cope better with mental illnesses, and that this claim is backed by research.
"Religious beliefs and practices of patients have long been thought to have a pathological basis and psychiatrists for over a century have understood them in this light. Recent research, however, has uncovered findings which suggest that to some patients religion may also be a resource that helps them to cope with the stress of their illness or with dismal life circumstances." - The Psychiatric Bulletin, June 2008
Both the Psychiatric Times and the Psychiatric Bulletin indicate that the idea that religiosity and mental health problems are correlated is a century-old and that is no longer believed in the psychiatric community. For Dr. Boghossian, who is a well-respected author and philosopher, to use speak with the same level of authority that he has earned in philosophy on a subject like mental health, for which he is not an expert, is unconscionable.
Not only was the assertion patently false, but it violates a skeptical method of inquiry that Dr. Boghossian lives and breathes. In his book, "A Manual for Creating Atheists", Boghossian repeatedly insists that philosophical skepticism is the only real player in the game of determining truth claims. I actually agree. The problem is that philosophical skepticism requires its adherents to follow evidence and adopt conclusions, even when those conclusions are in conflict with their personal feelings and preferences. Boghossian repeatedly says that he is not an antitheist in his book, but I disagree. It's clear to me that Boghossian doesn't like religion or religious belief. So much so that he's willing to make claims like "religious belief and poor mental health are correlated" when they are clearly not. Even a 15 minute search on Google will put that idea to rest.
I'm not necessarily trying to pick on Boghossian here. The point I am making is that he is clearly a very well-educated man who has made skepticism his life's passion. But he is human. Just like you and me. If a man who eats, breathes, and lives skepticism is willing to make such a baseless assertion, are we capable of doing the same? Yes. Of course. It's a condition of being human. So please be careful, oh atheist. And remember the next time you point a finger at someone for being delusional, there are three fingers pointing back at you.