This is incorrect.
First of all, it contradicts itself. It is itself a negative claim. If it were true, it would then be impossible to prove this negative claim, in which case it would be incorrect to argue that this particular negative claim is the only one that is proven to be true in all circumstances.
Secondly, we have real world examples. An alibi is a good example of proving a negative. The idea is to prove to the police that someone did not commit a crime. This is a negative claim. By providing evidence of being in another location at the same time, the person with the alibi has supported the negative claim that they did not commit the crime.
Third, we have examples of proven negative claims in the abstract, such as the law of non-contradiction. This law is a negative claim and can be proven logically.
Four, many (all?) negative claims can be rewritten as positive claims and vice versa.
Please stop using this meme as an actual argument, or more commonly, a reason to not provide evidence for a claim.
There is a crack in everything,
That's how the light gets in...