Just intonation is an unusual choice for Bach's works. I'm not aware of any evidence that he used it, and he certainly thought about tuning. His "Well Tempered Clavier" was meant as a demonstration of some unknown tuning system that was more flexible than the various meantone temperaments that were common in his time. But there are approximations of meantone using just intervals, so it's certainly possible that people have performed it that way.

But this sort of just intonation is not like barbershop, because all the intervals can be tolerably approximated with 12-equal temperament. The mood of the piece will change somewhat as some become more and some become less consonant, but it will remain essentially the same work. The barbershop seventh chord cannot be so approximated. The "ringing" sound of the seventh chord that defines the genre doesn't occur unless the seventh is tuned very noticeably flat from the closest 12-equal temperament equivalent. If you arpeggiate a barbershop seventh chord it sounds obviously microtonal.