Well, The Big Crunch has no real backing. It just is a possibility. The expansion of the universe is slowing down, but nobody knows what will happen when it stops. Will it just stay the same size? Or will it collapse in the Big Crunch? Or will time freeze? Or will time go in reverse, and we live our lives backwards??
As for the sixth-dimensional universe... Lets say that
if the Big Crunch happens (in our universe), It will most likely start as the universe getting smaller
veryslowly. All the matter in our universe will be condensed together. We would be crushed under immense pressure before the final implosion happens. Now, for the six-dimensional universe to 'Big Bang' it will need a
huge energy source. The energy could be provided by the final implosion of our universe. But we would all be dead by then. Even if the Big Bang in the sixth-dimensional univers occured now, it would still be along the lines of 300,000 years before it became anywhere near cold enough for solid matter to take form. And then, we would have to wait for everything to settle a bit, and for the suns to form. And then wait for the planets, and life to begin so we have a source of food.
As for us being even able to survive in a six-dimensional universe... We know of the four dimensions in our universe - Height, length, width and time. If these are four of the ten dimensions, can you even begin to imagine the other six? More than likely, we wouldn't be able to comprehend them and we would go insane.
Although, there is a theory that we are actually in a ten-dimensional universe now, and that there was ever only one universe. We just can't comprehend the other six dimensions.