In order not to get bogged down in disputes about the nature of rights or the nature of rationality in general, let me rephrase your question as follows: If someone in sound mind decides to end his or her life, should this be allowed? If not, why not?
One reason we might not allow a person of sound mind to commit suicide is that we think that person's decision is based on seriously incomplete or misleading evidence -- e.g. if her reading has led her to believe that her cancer is incurable when in fact it is quite easily eliminated. No matter how reasonable, and how well-read, a person is, it is possible to make bad decisions because one lacks good evidence. At the very least, we ought to intervene in such cases to make sure that the person has accurate information before acting. The very same evidence can lead different people to different conclusions, however, and we must not assume that everyone who disagrees with our own view (or an expert's view) is of unsound mind. Some people believe that a life without movement, or a life without language, is not worth living; others are confident that such a life is worth living; and neither group should be dismissed as irrational.
Another reason for disallowing suicide is the conviction that people do not own their lives and that no one should destroy what they do not own. This reason has been invoked by religious traditions that insist that lives belong to God and only God should be allowed to bring life to an end. But there are also less religious versions of this argument that view life as a part of nature that is not ours to destroy.
A third reason for disallowing suicide -- in certain situations, anyway -- concerns its likely effect on others. If a desperate mother's suicide is likely to wreck the lives of her children (leaving them in the hands of an abusive father, for example, or traumatizing them in such a way that they too will live desperately unhappy lives), then it may be right to prevent her suicide in order to save the lives of the children.
A more abstract reason for disallowing suicide concerns the apparent contradiction in the idea that we can improve a life by ending a life. The suicide's thought that she will be better off dead seems to contradict the fact that, if dead, she will not be anything. Her desire to retain control over her life by ending it in the way she wants to end seems to contradict the fact that there is no control over a life that has ended. There are other ways to express a suicidal intention, though, that do not lead to such contradictions.
I am convinced that there are many situations in which suicide is rational and allowable (situations of relentless pain, inevitable loss of mind, or endangerment of others). Furthermore, I think that suicide is something that we ought sometimes to facillitate (by supplying appropriate medications, for example). Because of the complexities described above, however, I do not think that discussion of this difficult topic is advanced by appeals to a 'right to suicide'.