I've gone through all of the NHL playoff series results at one time or another, and I can say pretty confidently that this has never happened.
Five has happened with some regularity. Off hand, both of the Oilers first round upsets in 1997 and 1998 saw the road team win five games.
I don't recall ever seeing six, at least in the 2-2-1-1-1 format. Back around war times, arena availablity (especially if the series involved the Rangers) sometimes saw a more obscure home-away format used. Perhaps six road wins might have happened in one of those series.