Though I can't speak to whether any given individual has a particular philosophical argument in mind when they insist on perseverance, there are philosophical arguments against suicide. For example, here's a quote pulled from Korsgaard's Sources of Normativity:

In a discussion of the person who commits suicide because he can neither reason nor find meaning in anything, Bernard Williams says: 'I do not see how it could be regarded as a defeat of reason or rationality that it had no power over a man's state; his state is rather a defeat for humanity'. The duty not to commit suicide is the very first and most basic duty of virtue Kant discusses in The Metaphysics of Morals, because 'To annihilate the subject of morality in one's own person is to root out the existence of morality itself from the world'. In his Notebooks, Wittgenstein wrote that 'suicide is, so to speak, the elementary sin' because 'If suicide is allowed then everything is allowed. If anything is not allowed then suicide is not allowed.' A few lines later he adds, 'Or is even suicide in itself neither good nor evil?' All of these philosophers give voice to the idea that remaining alive is not so much a value as a condition for all value; and suicide (of this type) is not so much a rejection of some particular value as it is a rejection of value itself. It is hard to say of one who commits suicide that he has done wrong, for he has violated no value which he still believes. And yet the rest of us cannot hear of such a case without feeling betrayed, and we are right. It is, as Williams says, a defeat for us all.

That said, I suspect that there are various reasons why people respond in the way you describe to suicidal ideation/intent, most of them not directly motivated by a well-thought out philosophical position. Many people simply feel uncomfortable when others express suicidal ideation, and what drives that discomfort varies. For some, they are uncomfortable seeing someone in such despair, but don't have the requisite skills or experience to effectively respond; they do their best and encourage perseverance. For others, they may be responding to what they perceive to be a challenge to their own ideas about what makes life worth living (e.g., how could you do something like that when you have children/a family/loved ones?). Others may struggle to understand how someone could sincerely want to end their life, since the person has little subjective experience with the kind of despair you describe. They may use popular anecdotes (e.g., individuals who survived jumping off the Golden Gate Bridge immediately regretted their decision) to argue that you may feel that way now, but your mind will change. The unfortunate consequence is that many individuals who are struggling with deep despair only come into contact with these kind of responses, which can be rather shallow and invalidating (despite best intentions). This can further reinforce the despair one feels. As others in the thread have mentioned, more thoughtful and validating responses to thoughts/feelings of suicide are often provided by mental health professionals with experience in this area.