I wish I were allowed to answer the answer before mine, but I can't yet, so I'll just incorporate it in this answer. The reason why British English absorbed the word from French surely does not date back to the Norman Invasion, as the fruit was known to Europeans only after the discovery of America. Whilst the family Cucurbitaceae was cultivated everywhere, the genus Cucurbita (mostly squashes) was cultivated predominantly in South America. Then the Europeans came to American and took it to the Old World with them, and in the late 19th century Italians in Lombardia grew (invented) them.
The first assumption of my pre-poster — that the Italian immigrants brought the term to the US — might be feasible but not probable. The Italians invented it to late. But it could be that they just got along with the name given to it by its inventors, as did some other languages including German.
As for the French term, I can only assume that it may be connected to the intellectual or academic proximity of England and France at that time and to French still retaining some of its "lingua franca" properties in Europe which it had prior to the 20th century. But that is just an assumption.