Note that as you accelerate a spaceship towards the speed of light you will blue shift the microwave radiation and increase it's temperature. Eventually you'll roast the ship and all aboard! This is actually happening for the Earth (well, not the roasting bit) as the microwave background is slightly hotter in one direction and cooler in the opposite direction. The effect is small, but it's actually a lot larger than the variations astronomers are trying to measure, so it has to be subtracted out.
Anyhow, back to the question, and I'd like to expand a bit on DVK's answer. It's common to think of the big crunch/bang happening at a point, so you could in principle navigate the spaceship away from that point. That is entirely wrong!
Imagine drawing out a grid with spacing of 1 light year. Although obviously we can't do this, you can easily imagine putting the Earth at (0, 0), Alpha Centauri at (4.37, 0) and plotting out all the stars on this grid. The key thing is that this grid is infinite i.e. there is no point where you can't extend the grid any further.
Now wind time back to 7 billion years after the big bang, i.e. about halfway back. Our grid now has a spacing of half a light year, but it's still infinite - there is still no edge to it.
Now wind back to 0.0000000001 seconds after the big bang. There's no special significance to that number; it's just meant to be extremely small. Our grid now has a very small spacing, but it's still infinite. No matter how close we get to the big bang we still have a infinite grid filling all of space. You may have heard pop science programmes describing the big bang as "happening everywhere" and this is what they mean. The universe didn't shrink down to a point at the big bang, it's just that the spacing between any two randomly selected spacetime points shrank down to zero.
In DVKs ballon analogy, you have to imagine shrinking the ballon down to a point of zero size, but all the surface of the ballon still exists.
Incidentally, if you try to calculate the size of the universe at the big bang itself you get zero times infinity i.e. zero spacing of the grid but it's still infinitely big. That's because the above is based on General Relativity and GR cannot describe what happened at the big bang itself. You just get sums like zero times infinity that don't make physical sense.
Gosh I've rambled on a bit! Anyway after all this my point is that the big crunch is approximately a time reversed big bang, so my argument above applies. The spaceship would get shrunk down to zero size and neither it nor it's passengers could possibly survive this.
Some notes: you'll have spotted that I casually stated the universe is infinte. By this it means it has no edge. Current observations suggest the universe is flat, i.e. not like a ballon, so having no edge means it goes on forever. The universe could be curved on a scale too large for us to measure. Either way we'll never know for sure.
Secondly, as DVK noted, it's widely believed that as you approach the big bang/crunch quantum gravity of some form will take over and the grid spacing wouldn't fall to zero. Both String and Loop Quantum Cosmology predict the spacing would decrease to some minimum size then bounce and start increasing again. Both theories are still speculative, but in any case it would make little difference to the spaceship since that minimum grid spacing would be around 10e-35 meters.
