Can't you prove a negative? It is a reversal of the burden of proof!
Despite the adage that holds the impossibility of proving a negative, it is
possible in some cases. However, the concept of gods is so diffuse that proving their non-existence is in my opinion a fool's errand.
I agree that the challenge to prove that gods don't exist is an attempt to reverse the burden of proof. But the sort of person who would issue that challenge would also never let logical fallacies get in the way of their march to triumph. On the other hand, if a person claims to know that gods do not exist, they take on their own burden of proof. Personally I don't make that claim. I don't believe that gods exist, and I mentioned the main reason for my lack of belief above. I think that it is reasonable to doubt the existence of gods, and for me that in itself is enough. If any of the gods that religions tell us about truly existed, the evidence for these beings would be unequivocal and no reasonable person would question their existence.
Then there is the fact that in the history of our species, thousands of gods have been worshipped. The attributes of this multitude of gods are contradictory--religions simply can't all be telling us about the same beings. The more likely explanation is that gods are a product of human imagination and desires, which vary by location and era; the gods reflect that variability.
One can entertain the thought that one particular god or one particular pantheon of gods does exist, but which? How to winnow out all the thousands of false gods? Being born into the faith of one's family is not evidence that their faith is the one true faith out of all available. Statistically, it's unlikely to be so.