The French bosquet is an Occitan (Provençal) or Catalan word (bosquet in both languages, "little forest") taked in the 16th century. Some scholars say that there is a confluence with the Italian boschetto (same meaning), too.
Regarding the surname Bosquet, etymologically, it is Occitan or Catalan, and very usual in some variants, in Catalan, for example: Bosquet, Busquet, Bosquets, Busquets. But in the registers, Occitan or Catalan people named Bosquet might appear as French people because a good part of Occitan and Catalan speaking territories are politically
France.
Same as with Basque surnames (Etxebarria/Etcheverry), but the Basque is not a Romanic language and there is not confusion with surnames of French origin. With the Occitan or the Catalan, the confusion is possible because all three languages are very close.