Bozon was born in St. Louis, Missouri, where his father, Philippe Bozon, played hockey for the St. Louis Blues of the National Hockey League (NHL), but was raised in France, where his father is from. Bozon then moved to Switzerland to further his career, before moving again to Canada after the Kamloops Blazers selected him in the 2011 CHL Import Draft. At the 2012 NHL Entry Draft, Bozon was selected 64th overall by the Canadiens. During his third season with the Blazers Bozon was traded to the Kootenay Ice. Internationally Bozon has played for the French national junior team in several tournaments and joined the French national senior team in the 2013 IIHF World Championships. Bozon contracted a life-threatening meningococcal infection in early 2014, though he ultimately recovered.