It is already implied in the “to claim YOU don’t exist” that YOU exist. You are the one making the claim that you do not exist. A hallucination isn’t a “you” but merely falsely appears to be a “you”.
Unless you believe your hallucinations are literal individual beings with their own identity, mind, consciousness thus ability to make claims on their own behalf? No, a hallucination is only the false appearance of a “you”, there is actually no “you” there at all.
If YOU don’t exist and are merely MY hallucination, then it is not the case that YOU are claiming anything.
If YOU are not claiming anything, then YOU are not claiming that you don’t exist.
But you do bring up an interesting point: how do we know that the people we are interacting with are not our own hallucinations? Well I say interesting but really it’s not that interesting, since radical skepticism is always competing with solipsism for the Most Idiotic Idea award.