I think what they're trying to get at is that one can actually take the whole of logic as an unprovable claim, and thus religion with it. In order to prove something, you must apply logic. If you apply logic to prove logic, you have a "begging the question" fallacy. Sometimes you will hear one of the clearer-thinking theists point this out. Literally, it does mean that there is a basic unprovable assumption in science. The theist arguer is setting a trap in this case, and often an atheist will bite the lure and try to deny that there is an assumption involved. Support for that stance, that there is no assumption, would require proof that doesn't rely on logic. Good luck there.
Really, when this comes up, it's necessary to admit that there is an assumption. The one assumption required, however, is still a lot smaller then the multiple assumptions made to support most religions.
Basing your evidence on logic alone does not make science. Or at least not within the current scientific paradigm. The position of using logic and reasoning alone to prove something is called
rationalism, and is largely derived from Descartes. (at least in it's modern form). This was challenged strongly by David Hume, who championed
empiricism. The difference, obviously, is that rationalism suggests that answers can only be found in rational logical thinking, empiricism suggests that the only way to arrive at truth is through observation and testing. Hume specifically added the viewpoint of extreme skepticism to empiricism.
Also, there is a difference between proof in the common sense, and proof in the philosophical sense. In philosophy of science, one cannot practically speaking prove a theory. Only disprove any argument against it. Quantum mechanics can calculate and predict results to a degree of accuracy which is comparable to measuring the exact width of the US to within the width of a hair.
The point is, in strict terms of philosophy of science, nothing can ever be proven, only disproved. In practical reality however, we can prove a vast amount of things to well beyond any reasonable doubt.
Also, the very act comparing empirically tested and validated knowledge with ontological or metaphysical deductive reasoning is engaging in a false equivalency fallacy. Especially considering the burden of proof. In scientific fields, it is incumbent upon the person making a claim to prove his hypothesis using inductive reasoning.