First of all, Kant's rationalistic counterargument is a little different than Hume's empiric counterargument as referenced here. Kant clearly identified analytic and synthetic versions of the ontological argument and criticized the analytic version as follows:
If the proposition is analytic, as the ontological argument takes it to be, then the statement would be true only because of the meaning given to the words. Kant claims that this is merely a tautology and cannot say anything about reality.
So for Kant in this a prior case, "the nonexistence of which" is just those which are not entailed by any tautology, thus no classic logical contradiction here. The ontic argument is simply a self-claimed tautology at best...
While empiricist Hume argued that nothing can be proven to exist materially using only a priori reasoning:
...there is an evident absurdity in pretending to demonstrate a matter of fact, or to prove it by any arguments a priori. Nothing is demonstrable, unless the contrary implies a contradiction. Nothing, that is distinctly conceivable, implies a contradiction. Whatever we conceive as existent, we can also conceive as non-existent. There is no being, therefore, whose non-existence implies a contradiction. Consequently there is no being, whose existence is demonstrable.
So clearly Hume's counterargument is similar to Popper's scientific empiric falsification principle, he regarded using a prior pure reason to prove some material existence fact is a category error. Since we cannot effectively falsify either such conceivable existence or conceivable nonexistence from our sense experiences, thus for Hume "the nonexistence of which" cannot imply contradiction either. But the reasoning is subtly different from Kant's above...