[3][4] Cozens initially attended Desert Mountain High School in Arizona, but was suspended midway through his junior year for an altercation with his baseball coach.

Baseball writer Keith Law reported that Cozens "shoved his coach and told him to 'f off' in the dugout during a game, so he was kicked off the team - the culmination of a series of incidents that turned many scouts off him, one even calling him a 'bully'".

[7] As a senior, Cozens posted a .520 batting average, and hit 19 home runs, in 33 games, breaking the school's single-season record, formerly held by longtime Chicago White Sox first baseman Paul Konerko.

[8] He ended the season by hitting a walk-off home run in the bottom of the seventh inning of the Division I state championship, winning the title.

[9] As a scout for the Philadelphia Phillies, Marti Wolever, among other representatives of the organization, evaluated Cozens before the 2012 MLB draft and watched him during a workout at Citizens Bank Park.

[12] "That was probably the guy I was most excited about", said Phillies general manager Rubén Amaro Jr., "Extraordinary power for a kid -- great body, runs well, seems to move pretty well".

[14] Cozens struck out 147 times in his first full season of professional baseball for the Lakewood BlueClaws of the Single–A South Atlantic League.

[16] The emergence of Rhys Hoskins and Cozens' high strikeout ratio dashed his opportunity to obtain a spot on the Phillies' Opening Day roster, in 2018.

[1][32] Cozens hit his first big league home run on June 7 against Chicago Cubs reliever Brandon Morrow.

On August 10, 2019, Cozens signed a two-year minor league contract with the Tampa Bay Rays and was assigned to the Triple–A Durham Bulls, and was placed back on the injured list.