We usually talk in terms of objects coming into existence. An object X, which was previously nonexistent, came into existence and now exists. This object could be anything, from a huge building to a subatomic particle. In such expressions we imply that existence is some sort of a quality that is had by X. In which case, to have existence as a quality, X would need to exist in the first place, and so it could not be called nonexistent prior to having existence as a quality.
Same goes for nonexistence. Because for Y to hold nonexistence as a quality, Y would need to exist and so could not be called nonexistent.
It follows that existence and nonexistence cannot be qualities, which implies that there cannot be any coming into existence from nonexistence.
P.S.: Inspired by the works of two great scholars of ancient India, Nagarjuna and Gaudapada.