I think the stems of vining squash will root at the nodes, but stems that are not prostrate are unlikely to do it.

The stems of seedling squash can be buried up to the cotyledons. They'll root from the stem.

Yes, for seedlings adventitious roots can form from the stems. They are called hypocotl roots. In mature plants, only from the nodes.

OP says she has seedlings with three inch stems which gave just got their second leaves.

More than 3 inches actually.

And they are prostrate at this point, since they all fell over - haha!

I will post a photo I promise.

Why don't you just plant them out deeply. They'll take care of themselves. Putting the seed end three inches deep isn't going to harm anyone, and you end up with a functionally erect plant.

Yes, just them in the ground.

Plant 'em deep. Job done.

Will do.

Thanks all!

You can put them in right up to the leaves.