In his second season with the Sarnia Sting of the Ontario Hockey League, Boucher broke the former 58 goals in a season record in the team held by NHL player Steven Stamkos with his fifth hat trick in the season against Saginaw Spirit. Boucher was rewarded for his outstanding play during the 2012–13 OHL season by being named to the OHL's First All-Star Team.
On March 13, 2013, the New Jersey Devils signed Boucher to a three-year entry-level contract. Reid scored one goal and had two assists in his first game with Albany in the 2012–13 season debut on March 30, 2013.
On December 4, 2013, Boucher was promoted to the Devils for the first time, following an injury to Ryan Carter. He got an assist and was plus 2 in his debut and became the first Devil to score in a shootout that season. Boucher scored his first NHL goal on December 18, 2013 against Robin Lehner of the Ottawa Senators in a 5–2 win.
In the 2016–17 season, having made the Devils opening season roster, Boucher was unable to make an impact in a depth role, appearing in just 9 games over the opening months of the season with the Devils.
On December 3, 2016, Boucher's 6-year tenure with the Devils ended as he was claimed off waivers by the Nashville Predators. He made his debut with Nashville the following day against the Philadelphia Flyers before he was sent on an conditioning assigned to affiliate, the Milwaukee Admirals on December 12, 2016.
On January 1, 2017, Boucher was placed on waivers by the Predators. The following day, on January 2, 2017, he was reclaimed by the Devils, but was again waived on January 3. On January 4, 2017, Boucher was claimed by the Vancouver Canucks. Boucher was scratched in all but one of the Canucks' games in January before drawing into the roster. He scored his first goal with Vancouver on February 28 against the Detroit Red Wings, scoring a late game-tying goal before the Canucks lost 3–2 in overtime. On March 25, he recorded his first multi-goal game; scoring twice in a 4–2 victory over the Minnesota Wild.
On July 24, 2017, the Canucks re-signed Boucher to a one-year, one-way contract worth $687,500, avoiding arbitration.