I had the same situation last year and tried to hand-pollinate with males from a different species. For me, it didn't work. The fruits initally looked like they were going to grow, but didn't mature and eventually rotted off the vine. I was trying c. moschata (Korean zuke) and c. maxima (kabocha). This year I have moschata and pepo. The pepo has male and female at the same time, but so far the moschata has been only male or only female, so I might try again - you never know. It's only July and our season goes into November so there's lots of time still for the vine to figure out itself how to reproduce.

According to always knowledgeable Google, some botanists say they can but very rarely while others say they cannot. Generally, if the squash varieties are of different species, then they cannot successfully cross pollinate.

Well, I have two small healthy-looking moschata fruit that were fertilized with pepo. Flowers long gone. Will see if they hold up.

I always thought if the fruit was pollinated, it would produce the fruit of the plant, but the seeds might be a cross or sterile. I'd expect the fruit will be fine, but if you saved the seeds, you'd get a cross, which would not be all that likely to be great.

None of the literature makes any distinction between pollination and fertilization. And they are two separate but related processes - pollination must happen before fertilization occurs. If the different species are unable to cross pollinate, are they still able to fertilize each other? Doesn't sound likely..