Upon graduating, he enrolled at Cornell University to play college ice hockey and was selected in the sixth round (174th overall) by the New York Rangers in the 2017 NHL entry draft.

[7] Opting to return to the NSMMHL for the 2014–15 season, Barron helped the team defeat the Moncton Flyers 4–3 to win the 2015 Major Midget Atlantic Championships.

[1] In 2016, as a Grade 11 student, Barron committed to playing NCAA Division 1 collegiate ice hockey at Cornell University.

[10] He had not been given an athletic scholarship and paid to enroll at Cornell, which head coach Mike Schafer explained was because "he wanted to combine that excellence in academics with a team where he felt he could get to the NHL and further his career.

[1] Barron was named ECAC Hockey Player of the Month for January after leading the league in scoring with 13 points in eight games.

[24] His dominating season also earned him a finalist position for ECAC Hockey Best Defensive Forward[21] and the team's Nicky Bawlf Award as MVP.

[25] During the offseason, he returned home to Halifax where he trained with fellow Nova Scotians Sidney Crosby, Nathan McKinnon, and Brad Marchand.

[28] Barron started his first season as captain by recording his first collegiate hat trick in the opening frame of a 6–2 win over the Yale Bulldogs, earning him Player of the Week accolades.

[31] He was also named one of 10 finalists for the Hobey Baker Memorial Award, given annually to the nation's top overall college hockey player,[33] and selected for the 2019-2020 CoSIDA Academic All-America Division I Men's At-Large team.

[40] During the first period of the Jets first game of the 2023 Stanley Cup playoffs, Barron fell onto Golden Knights goalie Laurent Brossoit's skate during a scrum in the crease, which led to 75 stitches above and to the side of his right eye.