I would not start with a contemporary socialist-influenced concept of “right” which means “the entitlement to have a thing”. I would also set aside the question of whether rights can be “proven”. Instead, I would start with the simple problem of moral evaluation: do you accept that there is a difference between “right” versus “wrong”? There is a good chance that your interlocutor does not believe in “right” vs. “wrong”, or “good” versus “evil”, so any discussion that presumes such a distinction when the other party rejects the distinction can go nowhere useful. You really have to start with the foundation, not the middle (where you pray that you’re standing on a solid foundation). Since you seem to be working towards a logical exposition of “rights”, I would suggest first getting a firm grasp of the concept “value”, because without it, all ethical discussions are pointless. Notice in The Virtue of Selfishness that Rand does not start with the concept of rights, she builds up to it. (So see also OPAR and Tara Smith Viable Values). Point #2 of 2046's answer, elaborated by Smith, should help get you past the "morgue-avoidance" thery of "living".
