This isn't true. The further away a galaxy is, the faster it recedes relative to us, and this can amplify well beyond the speed of light. This means different regions of space have had absolutely no communication with each other, as not even light could travel from one to the other to catch up with how they were receding from each other.
That's an interesting idea. My answer would be that even though every volume of space would be contracting if you reversed the clock, each of these volumes don't form their own singularities, instead the near-singularities coalesce at the point of the big bang.
http://en.wikipedia.org/wiki/Faster-than-light#Universal_exp...