Bourque was first given a chance to play with the Montreal Canadiens in 1938-39, thanks to a trade from the cross-town rivals, the Montreal Maroons. But unlike much of their glorious history, the Montreal Canadiens of the late 1930s were not a very good team.
That first season Bourque played in 25 games, where he posted a 7-13-5 record. Interestingly, Bourque apparently wore jersey #19.
Bourque's second season with the Habs was not a whole lot better. He was in the nets for 36 games, going 9-24-3. He was even loaned to the Detroit Red Wings for one game that same year, replacing the injured Tiny Thompson for a game on February 15th. Bourque was tagged with another loss there, falling by a 3-1 score against the New York Rangers.
In April 1940 the Canadiens trade Bourque to the New York Rangers, although he would never play in the National Hockey League again. He would star in the AHL for a couple of seasons before serving with the Royal Canadian Air Force in 1942. After his discharge from the war, he did not return to the nets.
Claude Bourque died in Ottawa, Ontario on May 13th, 1982. He was 67 years old.