Actually, doing a little digging it looks like courgettes (zucchini in the US) and marrows are in fact squashes, so we have lots of native squashes...

Courgettes and marrows (which are the same plant, just harvested at different stages) are the only squashes widely grown in the UK, but they aren't native; they come from the Americas. Butternut squashes are available everywhere but I don't think they're locally grown. Other vegetable squashes are very rarely seen.

Peter