I have had that same problem for years and finally just gave up on growing it altogether. I think the problem is the humidity causes the blooms to get moldy and then rot. I once read that it is best to get them up onto a trellis where they don't touch the dirt and where wind can keep down the molding and rotting. Hope someone else knows exactly what to do because I need to know also.

It's because they didn't get pollinated.....no really.......they can grow for a while if the female isn't pollinated but then they rot on the end and die.......weird eh?

Solution is get bees or hand pollinate.

Put some mulch under the plants to keep them off the dirt and google blossom end rot and see if thats what you have.

Spray with an organic anti-fungal?

I am growing 8-ball zucchini so practically the same thing. As stated earlier if the female blooms with the baby squash doesn't get pollinated then it drops off. Try to hand pollinate by taking one of long male stems without the baby squash at the bottom and rubbing the end of it inside the female bloom(Note that the female blooms opens in the early morning).
Otherwise, make sure to mulch and keep consistent watering not constant watering. Blossom end rot could be the problem, maybe you could add some bone meal or epsom salts for that.
My squash always does great until the vine borers gets them. I'm picking zucchini about every day right now.
Finally, and my best advice, is to visit the vegetable forum and do a search for the answer or ask a question there.

Pete

I roll pieces of pliable chicken wire into softball or soccer sized balls and place them underneath my squash plants. It allows air to circulate underneath the plants and keeps the squash off the ground. Try it!

Now if I would have caught the squash boring grub things in time, I'd be eating some squash :(

Also, not sure how long yours have been blooming, but I believe there are female and male blooms in the squash family. The females bloom first, and therefore, you might have some blooms and itty bitty non-pollinated squash succumb before the male blooms.

I'm having the same issue with my yellow squash and zukes. I have lots of ants crawling in the blooms but not doing damage so would think they are pollinating well. Could the ants be preventing pollination somehow?