If the intention is to talk about burden of proof, then that's what we should be talking about. That's clearly not the case because the parent comment replied asking how it was possible. They seemed to have meant it very literally that negatives are impossible to prove. It's a common saying and it's flatly wrong.

Additionally, in this case, the claim that a company can be trusted is much more difficult to prove than the claim that they cannot be. Burden of proof, difficulty of proof, and whether the claim is expressed as a positive or a negative have no intrinsic link.