Though Roy was born in Greenfield Park, Quebec, his path to the NHL went through the United States. He played two seasons of high school hockey at Deerfield Academy in Massachusetts, then had 104 points (54 goals, 50 assists) playing for Lincoln of the United States Hockey League. That led to him being taken by the Anaheim Ducks in the fourth round (No. 97) of the 2012 NHL Draft.
Roy enrolled in Northeastern University and was a prolific scorer during his four seasons with the Huskies, finishing with 150 points (65 goals, 85 assists) in 130 games and earning First-Team All-Star honors in Hockey East in 2014-15. He signed with the Ducks on April 8, 2016, after completing his senior season and spent 2016-17 with San Diego of the American Hockey League, finishing with 46 points (16 goals, 30 assists) in 67 games.