What would happen to Jupiter if a brown dwarf star collided with it, and vice versa what would happen to the brown dwarf star?
Brown dwarfs are heavier than Jupiter (jupiter has mass 0.000949 solar masses), although about the same size. Their compositions are fairly similar, with brown dwarfs depleted in deuterium if they are heavy enough (it is burned over a few million years after formation).
Adams & Laughlin modelled the collisions of brown dwarfs, mainly to estimate whether two dwarfs could merge to make a star heavier than 0.075 solar masses that would begin hydrogen fusion properly. They found that this could happen, and not too much mass appears lost as they splatter into each other if it is a somewhat glancing collision. Obviously, a near glancing hit might just scatter some of their atmospheres but not stop them.
Jupiter would likely behave the same. The result would be a heavier brown dwarf. It would be heated significantly by the energy of the merger, which in turn would make it expand to a larger size until it cooled down. Jupiter would also contribute deuterium so if the dwarf was heavy enough it would further heat up for a while. Given Jupiter's minimal mass compared to the 0.075 limit it is not likely it would be enough to turn the new object into a red dwarf.