Consider this example:
Skeptic: "You know what, I've never seen a ghost. I think that ghosts don't exist."
Believer: "You just made a claim. Burden of proof. Prove that ghosts don't exist!!"
But see, the skeptic did meet the burden. He has never seen a ghost. By default ghosts do not exist until evidence has been presented of their existence. This is why the burden of proof is always on the person with the positive claim. There is already plenty of evidence that something does *not* exist.
You're confusing two different claims. And this whole 'negative/positive' thing is nonsense. Those are asinine distinctions, and they're irrelevant to the 'burden of proof' as you see it.
The claim "I am skeptical that ghosts exist" isn't a truth claim (I guess it could be a truth claim about your thoughts, but that's obviously not what's at stake here). It bears no burden of proof. It's just an opinion or a state of mind.
The claim "Ghosts do not exist" is a truth claim and bears a burden of proof. It doesn't matter that if contains a negation. Check out the thread I started on this exact topic. It's trivially easy to prove a negative.
1. If ghosts exist, then irrefutable, reproducible evidence of ghosts exists.
2. Irrefutable, reproducible evidence of ghosts does not exist.
3. Therefore, Ghosts do not exist.
Regardless of its soundness, this proof is logically valid. If you want to assume that any statement featuring a negation is automatically true until proven otherwise, that's your prerogative. But you're going to run into some issues once you realize that every 'positive' claim can be reformulated into a 'negative' claim.
"It is not the case that ghosts do not exist" means the same thing as "Ghosts exist." Do you have to automatically assume the former since it has a negation in it? I mean, it has two of them. I guess we should doubly assume it to be true, yes?