I am not an expert but my cukes and squash and pumpkins did some growing. I also suffered some similar fates to you. From what I gather, A bag of fertilizer will not do the fertilizing you require. For that you need a nice wind, bees, birds, squirrels or a paintbrush. The "little" fruit you mentioned are not the set fruit created by a successfull pollination. These are the plant's attempt to go through with the process without the pollination and the eventual withering and drop off of the "hip" (for lack of a better term, like a rose hip) or tiny bud after the petals of the flower are gone.

I am not sure what creatures are in your environs but I plant complementary butterfly bushes and some other easy to grow flowers which bring bees to check out your flowers. You can check with your local gardner place as to what brings bees and birds and butterflies and they will all fertilize for you. You can also try the "paintbrush method" where you literally take a small clean watercolor brush or something like it and lightly tickle the insides of your male flowers and spread this on the available females when they appear. Be careful to not cross the varities if you intend to keep seeds for the next generation.
if anybody can correct any inaccuraccies please feel free. I can't think of the best methods and sometimes there are no "one size fits all" methods.

Doesn't sound like a pollination issue to me. What you have there is Blossom End Rot (BER). When you grow plants in pots and you have a situation where you let the soil dry out between waterings you can develop BER. You shouldn't have more than one summer squash in a pot if a pot is your only option, and anything less than a half whiskey barrel size and you'll be watering like crazy. You can do it in smaller containers you just need to keep in control of the moister issue with a timed drip system or something if you can't be around to water throughout the day.

I hate to sound discouraging, but I'm amazed that anyone ever manages to grow any kind of squash in a container -- considering that even the tamest of bush summer squash need from 4 to 9 square feet for a 3-plant hill in the garden.

When I've pulled SVB-killed summer squash out to replace them I've seen roots 5 and 6 feet long. And these were not full-grown but rather plants killed shortly after production began.

I grow zucchini in containers. Typically, the first flowers are all female, then sometimes I'll get mostly male, but eventually it evens out a little farther into the season. When the first male flower shows up, I use a small paintbrush or a Q-tip and hand pollinate whatever females are open. My containers are close to the building, so it's easy to keep an eye on them. There's quite a bit of info on hand pollination if you have some time to so a search.

Just getting started...
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Later on with fruit...
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Bush cukes in a similar box with a 3' teepee...
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These boxes are only 4"-5" deep and have a bottom. I the past I've grown Sugar Pie punks in a 2'x 2'x 18" box with a (strong)trellis attached and used nylon tulle to make slings for the fruit.HTH

Well which is more likely the culprit, blossom-end rot or poor pollination?

I thought blossom end rot was something that started out
as a small spot copper in color, then eventually becomes bigger and changes to black on the end of the fruit.

What ended up happening with my fruit is that I would have small, immature squash, that would just shrivel up and die instead of getting bigger. The type of blossom end rot I'm describing occurred with my tomatoes. Does blossom end rot
in squash display different symptoms?

When you say you've used 2' x 2' 18" boxes for pumpkins, could you use these same boxes for the bigger variety of squashes (butternut, acorn, buttercup, spaghetti etc.)

I was thinking of using 30 gallon containers for one zucchini plant but that sounds really big. Could I get away with using an 18 gallon container with 1 zucchini plant or would I have to use the 30 gallon container?

I would say lack of pollination caused your baby squash to shrivel.

Blossom-rot in tomatoes is caused by the inability of the plant to take up and utilize calcium in the soil, usually due to too cool temperatures, so it mainly happens to the first fruit of the season.

I only grew the smaller pumpkins in the box, so can't answer concerning the larger ones.

I would go with the smaller container for the zucchini.

For one thing plant more than one plant even if you don't need the extra zucchini. The chances of a male and female flower being open at the same time are twice as much with two plants then one, If bees won't do the pollinating for you then pick a male flower, remove the petals, and rub the center part on the center part of the female flower. Works better than a paint brush. And the male flowers usually die after one day anyway.
And as already posted, plant some plants like rosemary, lavender, or borage to attract bees to your yard, even some weeds like dandelions and oxalis will attract bees.