Maybe...I tried that last year with spaghetti squash, but they didn't germinate very well at all. I got only one good plant that made 2 squashes after climbing all over my bean trellis, after carefully cleaning and storing about 30 seeds. Most squashes grown commercially are in a giant field full of identical squash, so I would think that cross-pollination would not be so much an issue.
Well, my friend gave me some butternut squash seeds she saved from a store bought one last year. I planted a few this year and got several nice squash. We have cooked one and it tasted great. So I say it could not hurt to try.
There are 4 different categories of winter squashes. If these have grown next to each other, they may have cross germinated and produced fruit with seeds that are not true to type...
The cross pollination can grow some interesting fruits.lol
We had a compost squash plant that grew a unusual zuchinni/speghetti cross. The fruit looked somewhat like a zuchinni, staying green, but shaped more like a speghetti squash. The meat was much paler than a speghetti squash, more along the color of a zuc, but it cooked and tasted like a speghetti squash.
Not all the results of X-pollination will be so favourable. If you can get them to germinate, and have the garden space, give them a shot. You never know what you might come up with. All hybrids grown today are from X-pollinating experiments. Some never make it to the seed shelves, some do.