Gaunce grew up in Markham, Ontario, playing his minor hockey with the Ontario Minor Hockey Association (OMHA) Markham Waxers where he was a teammate of Steven Stamkos, Michael Del Zotto, and Cody Hodgson, before he was selected 50th overall by the Colorado Avalanche in the 2008 NHL Entry Draft. Gaunce played with the Mississauga St. Michael's Majors of the Ontario Hockey League (OHL) for three seasons, and on May 11, 2009, he signed a three-year entry-level contract with the Avalanche and attended the 2009 Avalanche training camp before returning for his final year with the Majors in the 2009–10 season.
In his first professional season in 2010–11, Gaunce was assigned to American Hockey League affiliate, the Lake Erie Monsters. After 54 games with the Monsters, Gaunce was recalled by Colorado and made his NHL debut in a 5–3 defeat against the Nashville Predators on February 12, 2011. In his third game with the Avalanche, he scored his first NHL goal and point against Marc-André Fleury and the Pittsburgh Penguins in a 3–2 loss on February 16.
During the 2012–13 season whilst entrenched with the Monsters and in the final year of his entry-level deal, Gaunce was traded by the Avalanche to the Dallas Stars in exchange for Tomáš Vincour on April 2, 2013. Gaunce was called up to the Stars on December 17, 2013.
Gaunce failed to make the Stars out of training camp in September 2014 and was assigned to their AHL affiliate, the Texas Stars.
On July 1, 2015, after he was not tendered a qualifying offer to remain with the Stars, Gaunce signed as a free agent to a one-year, two-way contract with the Florida Panthers. Gaunce was assigned to add leadership to AHL affiliate, the Portland Pirates, for the duration of the 2015–16 season. Gaunce responded in setting career best numbers offensively with 37 points in 75 games.
On July 1, 2016, Gaunce left the Panthers organization as a free agent, to sign a familiar one-year, two-way deal with the Pittsburgh Penguins. In the 2016–17 season, Gaunce expectedly began playing in the AHL with the Wilkes-Barre/Scranton Penguins. With Pittsburgh suffering a depleted blueline through injury, Gaunce was recalled and eclipsed his previous efforts in the NHL to record a career best 1 goal and 4 points in 12 games.
Gaunce for the third season in succession signed as a free agent on July 1, 2017, securing a two-year, two-way contract with the Columbus Blue Jackets.
His younger brother Brendan Gaunce currently plays as a left wing for the Vancouver Canucks. Brendan, who was made the 2nd overall pick by Belleville in the 2010 OHL Priority Selection, was selected 26th overall by the Canucks in the 2012 NHL Entry Draft.