Growing up in Burnaby, B.C., Nonis played junior hockey with the Burnaby Bluehawks of the BCJHL in 1982–83. He then began collegiate hockey in the NCAA at the University of Maine, playing 153 games from 1984 through 1988. Nonis spent his final two years for the Bears as team captain, leading Maine to back-to-back NCAA championship tournaments. After completing his Bachelor of Arts degree in 1988, Nonis played a year of professional hockey in Denmark and in 1989 returned to the University of Maine as an assistant coach for one season.