If you subscribe to classical logic (i.e., propositonal or first order logic) this is not true. Proof by contradiction is one of the more common classical logic inference rules that lets you prove negated statements and more specifically can be used to prove nonexistence statements in the first order case. People go so far as to call the proof by contradiction rule “not-introduction” because it allows you to prove negated things.
Here’s a wiki page that also disagrees and talks more specifically about this “principle”: source (note the seven separate sources on various logicians/philosophers rejecting this “principle” as well).
If you’re talking about some other system of logic or some particular existential claim (e.g. existence of god or something else), then I’ve got not clue. But this is definitely not a rule of classical logic.
I think you are assuming a level of competence from people that I don’t have faith people actually have. People absolutely can and do take “you cannot prove a negative” as a real logical rule in the literal negation sense. This isn’t colloquialism. This is people misunderstanding what the phrase means.
I have definitely had conversations with idiots that have taken this phrase to mean that you just literally cannot logically prove negated statements. Whether folks like you get that that is not what the phrase refers to is irrelevant to why I’m pointing out the distinction.