Veilleux was selected second overall by Shawinigan Cataractes in the 2009 QMJHL Entry Draft. 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.[2]
On March 6, 2013, the St. Louis Blues signed Veilleux to a three-year, entry-level contract.[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.
Veilleux was not tendered a qualifying offer at the conclusion of his contract with the Blues and remained an unsigned free agent heading into the 2016–17 season. On October 12, 2016, Veilleux agreed to a one-year AHL contract with the St. John's IceCaps, affiliate to the Montreal Canadiens.[4] In the 2016-17 season, Veilleux split the season between the IceCaps and secondary affiliate, the Brampton Beast. He featured in 53 games, adding to the forward depth of the IceCaps with 6 goals and 16 points.
As a free agent, Veilleux opted to remain within the Canadiens organization, signing a one-year AHL contract with new affiliate, the Laval Rocket, on July 20, 2017.[5] In the 2017–18 season, he appeared in a depth forward line role with the Rocket, contributing with 3 goals and 7 points in 52 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.[6] In the 2018–19 season, Veilleux made 50 appearances for the Americans in a checking-line role, notching 5 goals and 8 points.
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.[7]
On March 17, 2021, Veilleux was suspended two games by the AHL for using an obscene gesture during a game versus the Toronto Marlies.[8] He was suspended again a few games later following a knee-on-knee hit on Martin Pospisil.[9] Veilleux was once again suspended on May 6, 2021, a one-game ban for boarding.[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.[11]