The first thing to say is that the universe probably isn't headed for a Big Crunch. Rather the reverse in fact, as if dark energy exists it will probably keep the universe expanding faster and faster forever. There has been a suggestion from Andre Linde that the effect of dark energy may reverse and cause a contraction, but this is not something accepted by most cosmologists.
Anyhow, having got that out of the way, the next point is that a Big Crunch is not at all like a black hole. It isn't something that happens at a point in space, like a black hole, but instead it's something that happens everywhere. At the Big Crunch the spacing between every point in the universe decreases to zero. So there is no event horizon and no outside for any Hawking radiation to escape to.
It isn't possible to use General Relativity to calculate what would happen after a Big Crunch because at the moment of the crunch you end up with sums like zero times infinity, and these don't make mathematical sense. This probably just means General Relativity ceases to be a good description of the physics at a Big Crunch, but until we come up with a working theory of quantum gravity we aren't going to do any better.