You can cut half of the squash into quarter-size pieces and boil it until it's soft. Then, let it cool and do not put any salt of seasoning on it. It will be easy now to cut it further into smaller pieces. You can introduce a few pieces to mouse and see if he likes it. If he doesn't go for the pieces, try cutting the squash w/o the skin and mash it. Then, reintroduce it to mouse and see if he takes a bit. I think he will like it, but if not, then next time try buying baby food squash. He'll eat that, I'm sure.
Your mouse is young so I don't see a problem with steaming the squash until it is softer so that it is easier to eat. I have done both ways. Mine prefer it to be soft but they will eat it raw. Mine love squash.