Boston Celtics (2014–2017)
On June 26, 2014, Young was selected with the 17th overall pick in the 2014 NBA draft by the Boston Celtics. On July 10, 2014, he signed his rookie scale contract with the Celtics. Injuries forced Young to miss the summer league, most of training camp and part of the early season. Because of these setbacks, Young spent much of the season driving back and forth between Boston and Maine, playing for the Red Claws in order to receive consistent playing time. He had 11 D-League assignments in total during his rookie season. Playing in the D-League was something Young was originally against, but grew to love in time, even urging the Celtics to send him to Maine during the month of December. Young finally got an opportunity to shine for the Celtics on January 5, 2015, scoring a season-high 13 points in a loss to the Charlotte Hornets. In light of this performance, Young had a solid run of playing time in Brad Stevens' rotation, averaging 3.8 points in 12.9 minutes per game between January 5 and March 6. However, following that string of opportunities, Young managed just four more appearances to close out the regular season, and failed to appear in any of the Celtics' four playoff games against the Cleveland Cavaliers.
In 19 games for the Red Claws in 2014–15, Young averaged 20.7 points, 5.2 rebounds, 2.0 assists and 1.4 steals per game.
In July 2015, Young joined the Celtics for the 2015 NBA Summer League, where he averaged 9.4 points and 3.4 rebounds in five games. His performance was considered underwhelming, as he shot just 27.1% from the field and 22.7% from three-point range. He subsequently played the fewest preseason minutes of anyone who made the final 2015–16 opening night roster. Despite this, on October 30, the Celtics exercised their third-year team option on Young's rookie scale contract, extending the contract through the 2016–17 season. His early 2015–16 season opportunities mirrored his rookie season, as he spent five days with the Red Claws between November 3 and 9 on two different assignments before finally making his season debut for the Celtics on November 10, playing in the final 49 seconds of the team's 99–83 win over the Milwaukee Bucks. He received three more assignments to Maine following this. After playing in three games for the Celtics between November 24 and December 3, he almost got on a plane from San Antonio to Maine on December 4, but was informed not to minutes before boarding; the Celtics needed Young as insurance because of a quad injury to Avery Bradley. Young ultimately did not play for the Celtics against the Spurs on December 5. He went on to appear in seven of the team's next eight games, averaging 14.3 minutes per game over that stretch. Young received seven more assignments to the Red Claws in 2016 and appeared in three of the Celtics' six playoff games against the Atlanta Hawks.
In 16 games for the Red Claws in 2015–16, Young averaged 14.8 points, 5.6 rebounds, 1.9 assists and 1.0 steals per game.
In July 2016, Young re-joined the Celtics for the 2016 NBA Summer League, where he averaged 7.5 points and 2.8 rebounds in six games. On October 24, 2016, Young was retained by the Celtics for the 2016–17 season. On November 12, 2016, in the Celtics' ninth game of the season, Young scored a season-high 12 points in a 105–99 win over the Indiana Pacers. Young went unassigned in 2016–17 but only managed 29 games during the regular season.