Bourque is the 26-year-old son of Ray Bourque, the Hall of Fame defenceman who spent 21 seasons with the Bruins.
The deal was announced by Bruins general manager Peter Chiarelli on Saturday.
Bourque appeared in 73 games for Hershey of the AHL, and scored 27 goals. He was drafted 33rd overall by the Capitals in the 2004 NHL Entry Draft. In 2009-10, Bourque appeared in 20 games for the Pittsburgh Penguins.
I don't think today was my most solid game that I've had, some rebounds here and there. I know there's a lot of work to do, not every game is going to go as good as these last few. I've just got to make sure I keep working and keep trying to get better.
— Los Angeles Kings goaltender Martin Jones after stopping 38 of 39 shots in a 3-1 win over the Toronto Maple Leafs