I don't think it applies in the playoffs. Last night in the Hurricanes Bruins game, canes goalie left game early in the first period. The backup entered the game, and during the intermission they showed another canes goalie that had gotten dressed and was waiting in the locker room if needed. This goalie is a player for the canes AHL team, and had actually started a game for the canes earlier in the season. So it seems during the playoffs teams can provide their own team affiliated (professional) EBUG to dress if needed. I assume away teams travel with one as well.
The rules specify that the EBUG needs to be available for playoff games, though it does also say that if both goalies are injured any member of the player roster can dress as a goalie. And in the playoffs the roster limit becomes 50, so it's unlikely to come up unless all the other goalies under contract are also deep in the playoffs.
Waiting in the locker room is key though, you can only break the 20 players dressed rule if both goalies are injured.
