I'm a U.S. native speaker. It is a reference to the bitter, black-humor novel, "Catch-22" by Joseph Heller. It has entered the language. I think that in the U.S. it really is commonly understood. It's in the American Heritage dictionary, and I'll reproduce their definitions:
1. A situation in which a desired outcome or solution is impossible to attain because of a set of inherently contradictory rules or conditions
2. A contradictory or self-defeating course of action
3. A tricky or disadvantageous condition; a catch
In your case, the meaning that applies is meaning number 1, which is also the basic meaning in Heller's novel. In Heller's novel, Catch-22 was this: the only way you could get out of the military was to show that you were insane, but if you wanted to get out of the military it proved that you weren't insane--therefore there was actually no way to get out.