The concept of physically restraining someone from committing suicide is completely antithetical to the idea of man being free. You cannot believe in one while advocating for the other.
Suggesting that you put someone in a "rubber room" is a bit ridiculous, but it's normal to put someone that is suicidal under suicide watch. Keep in mind that mental health can be a huge factor in someone being suicidal. In other words, they're not thinking straight, so it's fair to help them during that time.
Having the right to end your life is different than others trying to help you through a challenging time and show you that there are other ways out.
Using the mental health angle to deny someone their most basic right is fraught with issues. Mental health is often defined in terms of conforming to the standard behavior under various measures. We define mentally unhealthy as sufficiently deviant behavior and use that as a rationale for restricting that behavior. This is a thinly veiled attempt at forcing conformity under a scientific premise. Unless one can be demonstrated to be out of touch with reality, we have no right to physically intervene with someone exercising their right to self-determination. If you believe this right has caveats, then you simply never believed in it to begin with.
That's just as difficult to define. My argument is that there's no black or white answer.
If someone is truly determined to end their life no human can stop them, only delay them for a little while.
Tell that to all the Bradley Mannings of the world.