This is a claim I've read so many times in comments that I think it deserves a little debunking. If you do a search on that little sentence, you'll see that it's very rarely if ever used in a scientific context but is repeated like a mantra by religious apologists. They seem to be persuaded that it is an established rule of logic.
Let's get it out of the way: it isn't. Here is a counter-example:
No even number that is larger than two is prime.
Done. I hope you'll agree that the proof to this is trivial.
One can prove a universal negative. Declaring otherwise pretty much constitutes a logical fallacy in itself. A universal negative poses no logical challenge whatsoever.
Now this of course applies to mathematics. When it comes to empirical truth, the challenge is quite different: one doesn't deal with proof whatsoever. Instead, you deal with evidence, no amount of which is ever equivalent to proof. Some assertions can be backed with more or less lines of evidence of various quality, which does make some assertions more valuable than others. But proof? Nope. Never. No big deal, too.
Apologists who resort to this false argument might as well say that science cannot prove anything. No it can't. Nor does it ever claim to.
But the absence of proof for not A does not mean that A, let alone B is true. The absence of proof for the non-existence of God does not mean that the Abrahamic God, or Zeus, or the Flying Spaghetti Monster exists. The absence of proof for the absence of a teapot orbiting the Sun between the Earth and Mars does not mean that there is a red teapot there.
Believe what you will. But if you are going to argue for your belief in scientific-sounding terms, or if you make claims that overlap with science (which all religions do), be prepared, and avoid making stuff up. Oh wait...