I'm sure the majority of the heat you receive is because your stance has absolutely no nuance to it. For instance, the majority of people who attempt suicide and live never attempt suicide again.

Similarly, suicide mostly seems to be impulsive -- if access to a certain method of attempting suicide is reduced, the corresponding decrease in suicide attempts via that method is not reflected in a corresponding rise in suicide attempts via other methods.

Lastly, suicide rarely occurs in the depths of depression. People who attempt to take their own life are much more likely to be heading into or out of a depressive episode.

While these aren't the only reasons that we should attempt to prevent suicides, I feel they provide sufficient counter-evidence to your argument. I will say that I am in favor of physician-assisted suicide, and there is definitely a contingent of people who are hell-bent on killing themselves, but for the most part most people who are suicidal will later be thankful to be alive.

If you'd like to know more, I suggest reading the article on suicide on the Stanford Encyclopedia of Philosophy:

http://plato.stanford.edu/entries/suicide/

Edit: Section 3.4 (Libertarian Views and the Right to Suicide) responds to your criticism very directly[1]

If you find that interesting (of if you still find yourself unable to sympathize with those who believe they should attempt to prevent suicides), I'd suggest taking a course on either biomedical ethics, or ethics in general. These topics have been debated for centuries at this point, and there are a number of very compelling arguments on both sides.

[1] This position is open to at least two objections. First, it does not seem to follow from having a right to life that a person has a right to death, i.e., a right to take her own life. Because others are morally prohibited from killing me, it does not follow that anyone else, including myself, is permitted to kill me. This conclusion is made stronger if the right to life is inalienable, since in order for me to kill myself, I must first renounce my inalienable right to life, which I cannot do (Feinberg 1978). It is at least possible that no one has the right to determine the circumstances of a person's death! Furthermore, as with the property-based argument, the right to self-determination is presumably circumscribed by the possibility of harm to others.