I assert that it is possible for someone to prove a negative. If I'm wrong, then the only way to establish this is to prove a negative. If I am right, you are wrong. Either way, you are wrong. Your argument is self-defeating. If it is impossible to prove a negative, then necessarily it cannot be that one can prove that no one can prove a negative (for, if someone could prove that it is not possible to prove a negative, they would be proving a negative).
Also, google "proof that the square root of 2 is irrational" and you will find a proof of a negative: there exists no number such that that number is rational and the square root of 2.
Proving that the square root of 2 is irrational is not proving a negative, . Proving that the square root of 2 doesn't exist would be though, which is not possible. Further, you state that "if [you] are wrong, the only way establish this is to prove a negative." This is certainly not the case, so let's dig into that example a bit with a hypothetical (any example can fit in here, though).
Let's say your position is that 1+1=3. I would prove you wrong, not by proving that 1+1 does not equal 3, I would merely need to prove that 1+1=2, precluding your argument from being correct. Thus, there would be no need to prove a negative in order to prove that you were wrong.
Here's a brief explanation about the logical falacy of a negative proof, fraudulently based on this concept:
A negative proof (classically knkown as appeal to ignorance) is a logical fallacy which takes the structure of: X is true because there is no proof that X is false. If the only evidence for something's existence is a lack of evidence for it not existing, then the default position is one of skepticism and not redulity. This type of negative proof is common in proofs of God''s existence or in pseudosciences where it is used to attempt to shift the burden of proof onto the skeptic rather than the proponent of the idea. The burden of proof is on the individual proposing existence, not the one questioning existence.
This is my argument here. It is ridiculous to even expect anyone to be able to prove that a supernatural entity doesn't exist. The burden is always on the one claiming existence. Don't you agree?