Sounds like the female flowers are not being fertilized.You need male and female flowers for pollination.You might not have male flowers yet.The male flowers look like a stem with a flower on the end.The female flower has a swollen base behind the flower--a baby squash.Be patient,the male flowers will come.

I hope you're right. Thanks so much.

Could also be blossom end rot. It is caused by not enough calcium in the soil. This could be due to overwatering, or could be the soil balance is off. You can easily check this with a soil tester, and if you do a search on Blossom End Rot ( or BER) on these forums you can get more information from those who know far more than I.
Good luck!
Meg

It probably is a pollination problem and could be that you simply don't have enough bees in the area. You could try your hand at pollinating them yourself by taking a q-tip and collecting pollen from the male blossom and transferring it to the female blossom. Both blossoms (male and female) need to be open for it to work good.