I have not read the book yet but it is already on my short list of titles to read. I think what we have here is a case of a very poorly constructed character arc. This is a problem with all the characters in the movie. We're introduced to each of the friends with a very short scene that essentially only tells us what they do for a living and what their special power is. Many of them are killed off before there is even a sense of why we should care about them as a viewer.
Why is Henry suicidal?
We don't exactly know but my interpretation is that he bears the burden of being able to hear peoples' thoughts. He has taken up the profession of being a shrink, but ultimately his gift has not given him any advantage to be able to help people. The example we're given in the film is the patient who is eating himself to death. Henry givens this patient the solution to his problem, but the patient ultimately gets defensive and runs out and we learn later that he ends up eating himself to death. I suspect this is a pattern for Henry. He is unable to use his power to help his patients and has to watch them destroy themselves.
Henry is still suicidal, but not enough to go through with it
Many suicidal people are not suicidal to the point of actually committing suicide. It is rather a cry for help. If you look at any depression quizzes online or at a doctor's office, they all include questions dealing with the level of suicidal tendency, including whether one has had thoughts of suicide, whether one has planned how they would do it, and whether one has actually carried out any plans. Henry has reached a point where he has acted on plans to commit suicide but is still hanging on.
When you say the character trait is never mentioned again in the film, this is not entirely true. There is a short scene meant to tell us this information that could be easily missed. After the car crash he is running through the snow talking to himself.
Henry: Let me get this straight. Yesterday you want to call it quits, throw in the towel, pull the plug. Today, you're begging, praying. Couple more hours. Couple more miles. Make up your mind.
He is depressed and suicidal but also conflicted.
His character arc is meant to show us his journey of conflict from wanting to end his life, and in the face of death and world destruction deciding that he wants to live after all.
As I said, though, the film does this very poorly, and not just with Henry. After I've read the book I will update my answer with any additional insights.