[2] As a freshman, Henry spent time playing offensive tackle, wide receiver, and defensive end for the football team.

He did not play tight end during his high school career, because Pulaski Academy's Spread Offense scheme did not feature that position.

Henry started his sophomore, junior, and senior years, winning a state championship with the Bruins in 2011.

Henry was named to the Parade All-America team for the 2012-13 school year and was one of the top recruits in the nation.

During a fourth-and-25 play in overtime against Ole Miss in the rivalry game, Henry caught a pass from quarterback Brandon Allen and upon seeing that he was going to be tackled short of the first down marker, Henry blindly flung the ball backwards as a lateral to running back Alex Collins.

[7][8] Henry skipped most physical drills at the NFL Combine but did compete in bench press with a position worst of 13 reps.[9] Henry managed to improve on his pro day by posting a 4.67 40 yard dash and completing 21 reps in the bench press establishing himself as one of the top tight ends in the 2016 NFL Draft.

[16] Henry entered training camp competing to be the backup tight end, against veterans Sean McGrath and Asante Cleveland.

Henry was named the Chargers' third tight end on their depth chart to begin the regular season, behind longtime Pro Bowl veteran Antonio Gates and Sean McGrath.

[17] In his NFL debut against the Kansas City Chiefs, Henry made one receptions for 20 yards in the season-opening overtime loss.

[20] On October 13, 2016, Henry led all Chargers in receptions, making six catches for 83 yards and a touchdown in a 21–13 victory over the Denver Broncos.

[27] On May 22, 2018, Henry suffered a torn ACL during organized team activities, and was ruled out for the entire season.

[28] On September 1, 2018, he was placed on the physically unable to perform list, giving Henry a chance to return later in the season.

[29] He was added to the active roster on January 7, 2019, ahead of their Divisional Round game, which they lost to the eventual Super Bowl LIII champion New England Patriots.

[41] Over the first 10 games, Henry caught seven touchdown passes, establishing himself as a frequent red zone target for rookie quarterback Mac Jones.