This is a very common problem with squash in my area and is simply because the female flower on the squash plant is not being pollinated. It looks very very similar to blossom end rot, a calcium deficiency, but no amount of calcium fertilizer will do anything to fertilize a flower, which is easy to do yourself. This has the information you need that explains how to handle this problem. It also usually an issue early in the season, and for the first several baby squash. http://blog.mountain-plover.com/2009/08/03/hand-pollinating-zucchini/. Adding calcium to vegetable gardens actually is a good plan before planting, tomatoes especially get blossom end rot, and calcium is the only thing that will correct that.
Blossom end rot affects squash in addition to tomatoes. It's possible that what you have. If so, it's a result of a calcium deficiency. There are suggestions here from the Clemson extension service:
I agree with Douglas, probably blossom end rot. A lot of veggies get it. Add a calcium fertilizer. For the long term, add some ground eggshells.