As a youth, Bourque played in the 2003 Quebec International Pee-Wee Hockey Tournament with the Boston Top Gun minor ice hockey team. After playing high school hockey at the Cushing Academy, Bourque decided to follow in his father's footsteps and play junior hockey in the Quebec Major Junior Hockey League rather than play college hockey at the University of New Hampshire, whereas his brother Chris decided to play for Boston University. He was drafted in the 7th round by the Quebec Remparts. Playing with Quebec gave Bourque the opportunity to play for his father's former teammate and fellow Hockey Hall of FamerPatrick Roy, who was the Remparts' coach. With Quebec in 2009–10, Bourque scored 19 goals and had 24 assists for a total of 43 points in 44 games. His 43 points ranked 10th among QMJHL rookies, even though he missed more than 20 games to injuries and international play. He added another 10 points in 9 playoff games. In 2010–11, he scored 26 goals and had 33 assists for 59 points, which ranked 38th overall in the QMJHL, despite only playing 49 games. His 59 regular season points ranked 3rd on the Remparts, and he also ranked 3rd on the team in goals and 5th in assists, even though he missed 19 games. In 18 2011 QMJHL playoff games, Bourque collected 16 points, tying for 13th in the league as of the end of the semi-finals. On March 16, 2011, Bourque signed a three-year professional entry-level contract with the New York Rangers. Bourque was expected to join the Rangers' American Hockey League affiliate, the Connecticut Whale, for the playoffs after the Remparts' playoff run ended. However, the Whale's playoffs ended prior to the Remparts'.
Professional
Bourque had an impressive 2011 training camp with the Rangers, leading to his inclusion on the Rangers as the team traveled to Europe for its last exhibition games and the start of the regular season. However, Bourque was assigned to Connecticut before the start of the season and began the 2011–12 season with the Whale. He scored his first professional goal for the Whale on November 4, 2011 against the St. John's IceCaps. After the 2012–13 Whale season, he was added to the Rangers' playoff roster. On August 8, 2014, the Rangers announced they had agreed on terms of a contract with Bourque, who was a restricted free agent at the time. The contract is for two years, with an average annual value of $562,500. He was promoted to the Rangers on April 9, 2015 and made his NHL debut that night against the Ottawa Senators. He did not score but registered one shot on goal and one takeaway. He returned to Hartford on April 12 and then was recalled to the Rangers during the 2015 Stanley Cup playoffs on May 2. During the 2015–16 season at the NHL trade deadline, Bourque was traded by the Rangers to the Washington Capitals in exchange for Chris Brown on February 28, 2016. He was directly assigned to AHL affiliate, the Hershey Bears, marking a reunion with older brother Chris. After a run to the Calder Cup finals with the Bears, Bourque opted to continue in Hershey agreeing to a one-year AHL contract as a free agent on July 6, 2016. As an unsigned free agent from the Bears after parts of two seasons, Bourque was unable to secure a contract over the summer. He accepted an AHL try-out contract to attend the Bridgeport Sound Tigers training camp on September 24, 2017. After a successful tryout, Bourque signed a one-year AHL contract to remain with the Sound Tigers on September 29. During his third year with the Sound Tigers in 2019–20 season, Bourque contributed with 11 points in 49 games before he was traded by Bridgeport to the Charlotte Checkers in exchange for Terry Broadhurst and Cedric Lacroix on March 2, 2020.
International play
A dual citizen of the United States and Canada, Bourque chose to represent the United States in international hockey play. Bourque won a bronze medal in the 2008 IIHF World U18 Championships with Team USA and the following year was a member of Team USA's gold medal winning 2009 World U18 Championship team. He scored two goals and had three assists for the 2008 bronze medal winners. He scored 1 goal and had 6 assists in 7 games for the 2009 gold medal winners, and led the team with 32 shots on goal. The following year, he was a member of the USA's 2010 gold-medal World Junior Championship team. He scored no goals but earned three assists during the tournament. His teammates on the gold medal winners included fellow Rangers' draft picks Derek Stepan and Chris Kreider. The next year, he was an alternate captain for the USA's bronze-medal-winning 2011 World Junior Championship team. He once again earned three assists in the tournament. One of his assists came on the game-winning goal in the bronze-medal game. He was named Team USA's best player in their win over Switzerland. He also tied Chris Kreider for the 2nd on the team in shots on goal with 20.
