I've heard it with varying degrees of the R sound. There's a common shorthand "bougie" (BOO-zhee) that people often hear before learning the original term, so they'll maintain the pronunciation into BOO-zhwa.

Sometimes the R is slightly swallowed so it sounds more like BOH-zhwa, maybe very light throat vocalization. Or people skip over it and it's buh-ZHWA. Some commit fully for BOR-zhwa.

Universally seems to maintain (my non-native understanding of) the French "oi" and silent S.

I have yet to hear anyone pronounce it correctly: bor-gee-oice.