The proposition that "all swans are white" is easily disproven by even a single instance of a black swan.
However, given that some birds are blue, some overimaginative ornithologist might declare, "some swans are blue", or, even more difficult to disavow, "blue swans formerly existed."
To demonstrate – prove – this blue-swan proposition would require finding a blue swan. However, how could one disprove the existence of something that does not exist? Even though a survey of gazillions of swans would (almost certainly) only turn up white and black swans, it is remotely possible that there could have been a blue swan that left no trace of its existence. Vanishingly unlikely, of course, but remotely possible.
It is not logically possible to categorically demonstrate such a non-example. Such a failure of disproof would not indicate that the proposition had any validity, because it is even less likely that a blue swan has ever existed than that my hypothetical demented ornithologist should be correct.
When religionists taunt, "it is up to atheists to prove that God does not exist," they are making an illogical demand on several counts: a demand for the logically impossible, a shift in the burden of proof, a fallacious argument from ignorance, and ignoring or denying the counter-evidence (refutation of apologetics, scientific explanations).
Theistic claims fall into the "some swans are blue" category. Claims for the existence of some supernatural entity that purportedly once interfered with the cosmos (deism) fit into the "blue swans formerly existed" category. However, the claimed theistic supernatural supposedly interfered with the universe post-Bang and there ought to be some physical evidence that could be explained only by reference to this purported interference. Just like the case for blue swans, there ought to be some evidence of the purported existence. That is, there cannot be two contradictory explanations for one event.
Apologetic claims for God's existence fall into the same category as the swan feather that I altered to make it appear blue. The fact that I invented the blue-swan example purely for the sake of argument indicates that we can make assumptions about the possible truth (or untruth) value of a proposition by examining its source. The fact that creation myths and deities have been invented by almost every civilization yet examined, suggests that, like the blue-swan, we can dismiss claims for supernatural deities along with discarded mythologies and Russell's teapot.