Kozun began his junior career with the Calgary Royals of the Alberta Junior Hockey League in 2006–07. He also appeared in 11 games with the Calgary Hitmen, scoring one goal. Kozun moved up to the Western Hockey League full-time in 2007–08 as he recorded 19 goals and 53 points in 67 games for the Hitmen. He improved to 108 points in 2008–09, second in the League behind Casey Pierro-Zabotel's 115, and was named a First-Team All-Star by the WHL. His season also earned the attention of scouts. The diminutive forward relies on his speed and skating ability and was praised by Hitmen Head Coach Mike Williamson for his competitive nature and willing to play any role the team asks of him. Williamson added that his sense of anticipation allows Kozun to see plays on the ice develop before they happen, stating it to be a trait found in elite athletes. His small size, standing, initially led WHL teams to overlook him, as well as NHL scouts who argued he was too small for the NHL. He was left unranked by NHL Central Scouting in his first year of draft eligibility in 2008, and went undrafted that year. Following his 108-point season with the Hitmen, the Los Angeles Kings selected Kozun with their sixth-round pick, 179th overall, at the 2009 NHL Entry Draft. Kozun led the WHL in scoring in 2009–10, scoring 13 points in his final five games to finish with 107 points to win the Bob Clarke Trophy.
Professional
The Kings assigned Kozun to their American Hockey League affiliate, the Manchester Monarchs, to begin the 2010–11 season. In his home debut for the Monarchs, on October 16, 2010, against the Providence Bruins, Kozun scored two goals, the first of which was a penalty shot goal at 19:26 of the game's first period. The goal was the first of two penalty shot attempts by the Monarchs, a franchise record for penalty shots awarded in a game. After four seasons in the Kings organization, Kozun was traded to the Toronto Maple Leafs on January 22, 2014, in exchange for forward Andrew Crescenzi. Kozun was assigned to the AHL's Toronto Marlies. After an impressive display in pre-season Kozun made the final cut for the Maple Leafs for the 2014–15 season. He scored his first NHL point with an assist on a goal by Nazem Kadri in the season opener against the Montreal Canadiens on October 8, 2014. He was later placed on waivers on December 17, 2014. He would remain with the organization, however, and later in the season, on February 20, 2015, Kozun scored his first career NHL goal in his eighth game played. On June 18, 2015, Kozun left the Maple Leafs to sign a one-year deal with Jokerit of the Kontinental Hockey League. As the League's only Finland-based team, Kozun made an immediate impact with Jokerit in the 2015–16 season. He led the club with 34 assists and 49 points in 58 games to help Jokerit claim the Bobrov Division before suffering a first round defeat to the hands of Torpedo Nizhny Novgorod. After three seasons with Lokomotiv Yaroslavl, Kozun left as a free agent following the 2018–19 season. He signed a one-year contract to continue in the KHL with Metallurg Magnitogorsk on May 1, 2019.
International play
Kozun earned the attention of both the Canadian and American national junior teams whilst leading the WHL in scoring early in the 2009–10 season. Kozun was reportedly guaranteed a spot on the American team at the 2010 World Junior Ice Hockey Championships, but declined as he wanted to show loyalty to the nation that developed him as a hockey player. He later earned a spot on the Canadian team, his first time representing Hockey Canada internationally. During round robin play, he scored the shootout winner to cap a dramatic 5–4 come from behind victory over the Americans, leading Canada to first place in their pool, and a bye into the semi-final round of the tournament. Canada, however, lost a rematch against the Americans in the gold medal game, also in overtime, to settle for the silver medal. Kozun was selected to play for Team Canada at the 2018 Winter Olympics where he helped them win a bronze medal.
