We know that Seven of Nine had dreams, but she wasn't fully Borg. I'm curious whether anything in the canon answers the question of whether Borgs dream while regenerating.
It depends on what you mean by dream, and who you want to include as Borg.
From the TV shows, it would seem that your regular Borg drone would not dream in the sense of what you or I would dream - they are always part of the collective, and even when regenerating they are sensitive to the voices of the other Borg speaking in unison.
We do see some irregular cases, though - Borg that become separated from the collective, like the Unimatrix Zero Borgs, or Seven of Nine. And we certainly see Seven dreaming in at least a couple of episodes.