Let's say that it's scientifically proven that ghosts exist. Would they then stop being supernatural and become natural, thus making it impossible to ever have proof of the supernatural?
Only in a sense. Only if they are “supernatural” in the sense of going beyond one’s current conception of “natural”.
For example, someone photographing a nuclear explosion would be “proving the existence of a supernatural entity” if they showed that photograph to someone whose physics hadn’t yet gotten to the point of understanding nuclear fission yet.
Nuclear fission is supernatural to previous-generation physics.
It’s not supernatural to Real Physics but Real Physics is an ideal that will always elude us.