Certified GKH Gardening Expert
Yes, it is possible. Nearly any plant can be grown hydroponically. The issue is that squash plants get very large, even bush varieties and it can be quite difficult for the home hydroponic gardener to accommodate their size and support their size on typically elevated hydrponic beds.
If you are going to try growing these, I would recommend that you grow vining varieties as these can be coaxed up strings or wires and would make it easier to accommodate their size.
That being said, here is an interesting video from a gentleman who is growing bush varieties hydroponically: