Butternut squash are harder to grow than other kinds as they generally need more heat and a longer growing season. You have done the right thing getting an F1 hybrid increasing your chance of success.
They belong to the cucurbita moschata group and will not pollinate with other kinds of squash. (c. pepo, c.maxima and c.mixta all pollinate with one another but not c. moschata ) So unless someone else is growing butternuts nearby the bees only have your three plants to work on. This however should be plenty and the fruit seem to be being pollinated to some degree.
So perhaps the recent cooler weather with rain does not suit them. In other words they need more heat. Some of my onion squash (uchiki kuri) always seem to go brown and fall off so it's not just you.
I suspect over the next week or so proper fruits will form. The question then will be is their enough time left or them to mature properly.
The only moschata I grew was a cheese type called musquee de provence, (120 days from planting to harvest) good looking but watery flesh inside. I started the seeds two weeks earlier than other squash late march and later planted them out under a cloche for a couple of weeks to increase the chance of success.
The other squash had no such mollycoddling.
One good thing about moschata is that it seems resistant to powdery mildew that plagues other squash.
