Greig played major junior hockey with the Brandon Wheat Kings of the Western Hockey League (WHL). After his selection by the Ottawa Senators in the 2020 NHL Entry Draft, Greig was signed by the Ottawa Senators to a three-year, entry-level contract on December 30, 2020.[2] Greig played two more seasons in the WHL, before joining the Senators' American Hockey League (AHL) affiliate, the Belleville Senators, at the end of the 2021–22 season and played in Belleville's Calder Cup playoff loss to the Rochester Americans.[3] He joined Belleville full-time at the beginning of the 2022–23 season. Greig was recalled by Ottawa on January 23, 2023.[4] Greig made his NHL debut on January 25, 2023, in Ottawa against the New York Islanders playing on a line with Alex DeBrincat and Claude Giroux. He registered an assist in the game.[5] On February 19, 2023, Greig scored his first NHL goal against the St. Louis Blues.[6] Greig was assigned to Belleville on February 25, 2023, after playing in 11 games for Ottawa.[7] He was recalled again by Ottawa on March 15, 2023, after Ottawa had lost four straight games.[8]
Greig was selected for Team Canada's roster for the 2022 World Junior Ice Hockey Championships. He distinguished himself in the early going, being named the team's best player in two of four group-stage games.[9][10][11] He then suffered an injury in the first period of the quarter-final game against Team Switzerland, as a result of which he missed the remainder of the tournament, considered a significant loss for the team.[12] However, Greig shared in Team Canada's eventual gold medal win.[13]