On both my summer and winter squash, I am seeing young squash, but before they are even an inch big, they get soggy and fall off.
Is this common and how can I correct the problem so that my squash will fully mature and ripen?
sounds like there not getting pollinated.
If they are only an inch I doubt they are even flowering. Sounds like your plant is stressed, it is common for small fruit to abort when the plant is not capable of supporting fruit. This could have many causes, lack or excess of nutrients or water, other large fruit already on the plant, or not enough sun, to name some common ones. A picture would be helpful but aborting small fruit is not a cause for concern the plant simply can't support each and every fruit.