He led the Cataractes to win the 2012 Memorial Cup[1] before being traded, prior to the start of the 2012–13 season, to the Moncton Wildcats in exchange for defenceman Brandon Gormley.

[3] For the duration of his rookie contract with the Blues, Veilleux remained primarily with AHL affiliate's, the Peoria Rivermen and Chicago Wolves, appearing in 167 games.

After two seasons within the Canadiens organization, Veilleux left as a free agent and later agreed to a one-year AHL contract with the Rochester Americans, affiliate to the Buffalo Sabres, on August 3, 2018.

Having played the last five seasons primarily in the AHL, Veilleux as a free agent opted to return to the Kalamazoo Wings of the ECHL, signing a one-year contract on October 1, 2019.

[10] Following two productive seasons with the Rocket, Veilleux left North America as a free agent, signing his first contract abroad with German club, Eisbären Berlin of the DEL, on June 29, 2021.