Good afternoon,  Sorry to hear about your pumpkin fruit loss. If possible, could you submit a photo of your plant, the stem, blossom, and fruit? This would help me to evaluate your problem and provide recommendations. Thanks.

Hi Gail!
As you can see, the plants themselves are doing well. I've included a pick of one blossom with the bulge developing underneath. I will bet you when I go out tomorrow, there'll be nothing but a lopped off stem (which I hope you can see in the middle of the third pic). 
-Serena

Hi Serena, 
After reviewing your photos, I don't think you have a problem with squirrels. Based on your photos, I think your plants might be experience a few different issues. There are a few reason for pumpkin fruit drop. One, pollination problems, that's the most common. If you have problems with pollination, even if tiny fruits begin to grow, the little fruit will soon wither and fall off the vine. Two, fertilizer issues. Too much nitrogen fertilizer can put baby pumpkins at risk. After the plant is established you should use fertilizer with a low nitrogen ratio. Three, stress, excessive humidity, and high heat can cause early fruit drop. Proper fertilization and proper irrigation to keep soil cool and moist, and applying mulch can help cut back on heat stress. Four, blossom end rot can also cause early fruit drop, although, if you do not have the fruit exhibiting a black spot on the blossom end, this might not be your case. Too much nitrogen fertilizer will tie up calcium and cause blossom end rot, if you have black spots at the end of any of your pumpkins, stop applying nitrogen fertilizer.

So my recommendations are, apply some compost to your plants, allowing them to reach for the nutrients needed from the soil, cut back on nitrogen fertilizer, mulch all of your plants well to help hold on to soil moisture during excessive heat, use a soaker hose for irrigation or water well and deeply, about an inch a week and check for proper pollination. Plant some flowers in that space that will attact pollinators, or consider hand pollinating a few blossoms.The first flush of blossoms are male, followed by female, they can be identified by their stamens. You can remove one or two and rub all over the female blossoms if you want to give that a try.

Ok- yeah, we have been watering but have let the soil dry out at times. Will look into fortifying that. Haven't fertilized at all.
So, the stalks that once had flowers and look like they've been cut off at the top are of no concern?

Here is another option if you haven’t been using fertilizer at all. It could be cut worms. Here are two photos of the grub that comes from the soil and cuts off stalk, usually in evening and the moth that lays the eggs. Scout for these pests. If you see any digging around the plant you can remove and kill, and place paper “collars” around your stems.

Aha!
Ok- will definitely go look to see if there are any of those around!
Thanks so much!!

I hope all this information is helpful.