The term "vegetable" is a culinary term, and squash is prepared like a vegetable. For another example, tomatoes are fruits but are prepared like vegetables. Squash and tomatoes can be both fruits AND vegetables. This is my position on the "is X a fruit or vegetable?" issue.

I mean, the idea of a "vegetable" isn't a well defined group of plant parts like fruits are. Vegetables are a mix of seeds, roots, leaves, stems, etc. all of which are quite different. It's just "parts of a plant that can be cooked as part of a meal":

"a usually herbaceous plant (such as the cabbage, bean, or potato) grown for an edible part that is usually eaten as part of a meal also : such an edible part" according to the Merriam-Webster dictionary https://www.merriam-webster.com/dictionary/vegetable (similar definitions exist for other dictionaries, some highlight that vegetables are usually used to make non-sweet dishes)

The TLDR is that vegetables are loosely defined as "plant parts that are used to prepare meals, usually non-sweet dishes" and is a culinary term rather than a botanical one like fruits can be. So an item (like tomatoes or squash) can be both a vegetable and a fruit, the former culinary and the latter botanically. They aren't mutually exclusive.

Depends on which classification system you use. Botaically it is a fruit. But culinarily it is a vegetable.