To get started we need some context. The term supernatural is terribly nebulous. Most would probably use the term to describe unicorns, god, or psychic powers. Let's get a more concrete definition in place. Here is one of the definitions Sean Carroll has provided:
The lawless: things that affect the world in ways that are observable (directly or otherwise), but not subject to the regularities of natural law.I feel pretty comfortable with that definition. Basically, something that is supernatural would have to be completely unpredictable. Anything that is natural is predictable and anything that acts predictably fails within the domain of science. That's what science does. It makes predictions about the natural world tests them.
So how do you prove that something has unpredictable behavior? I'm not sure that's possible. The mere fact that we would try to prove it indicates that we are expecting predictable behavior. The logic feels circular and wrong. This makes the idea of lawlessness seam almost nonsensical.
My point is that anything that operates using the laws of cause and effect is natural and falls within the domain of science. Even if this thing is a super powerful being that shoots lasers from its eyes and raises the dead, it is still natural if its behavior can be predicted. The only definition I could accept for a deity is a being that truly had free will. This being would have behavior that is impossible to predict, and not just because it would be difficult.
If your idea of a god falls outside of that category, into the range of predictable, then you have a lot better chance of someday being able to prove it to me. But then if you did that, it wouldn't really be a supernatural god anymore would it?
So now we come back to the original question. Can there be proof for a supernatural god? Logically, I don't think so. Is it even a good question? Not really. Is it possible that a supernatural being exists? Maybe, but I doubt it could be satisfactorily proven.
I find it comical that religious folk spend so much time trying to prove that their deity exists. If they ever succeeded it would seem that the deity isn't actually all that powerful. It was predicted and proven using the scientific method? Seems sort of weak.