Squash are heavy calcium feeders same as all melons and tomatoes and will suffer from blossom end rot which is exactly what you are describing. To correct add extra calcium at planting time by using bonemeal or gypsum. Not Keeping the ground moist is another problem dry soil stresses the plants and will cause fruit to abort and affects the plants ability to take up calcium. So keep well watered especially in this heat and add calcium at planting. There are products for blossom end rot that can be used on growing plants check your local garden center or feed store and ask.Rodger

I don't know Rodger, you always have great advice, I use it alot, but are you sure you're not overthinking this one? It sounds to me like dirtnapper has a pollination problem. Dirtnapper, have you tried hand pollinating your squash? It's likely that you don't have enough bees and they need some help. Take the male blossoms (the ones with no baby squash on them) and peal off the petals. Then take the naked stamen on a stem and swish it around into the open female blossom (one with the baby squash attached.). If the blossom has recently closed, you can still peal back one petal and get the job done. Voila! You are a bee, and have pollinated your squash. You'll know it took if the baby squash is twice the size the next day. If it's not, the pollination didn't take and probably that baby squash will grow soft and rot starting at the end closest to the flower.

Hope this helps!
Susan