In his junior year, the team tied Governor Thomas Johnson High School for the league championship.

South Carroll then lost in the regional semi-finals to top-ranked Paint Branch High School.

After graduating, he attended West Nottingham Academy (a small college prep school) in Colora, Maryland, averaging 28 points, 16 rebounds and 9.7 blocks per game.

Boone played college basketball for UConn and was named to the Big East All-Rookie Team.

Set a Big East Tourney record for rebounds in a game by a freshman with a career-high[citation needed] 16 boards in quarterfinal win at Notre Dame.

After the summer league he underwent surgery to repair a torn labrum in his left shoulder and was expected to be out for 4–6 months, however he returned much earlier.

Suffering the effects of the injury and missing the pre-season training, Boone had an inconsistent but promising rookie season.

The following game, on March 28 against the Indiana Pacers, Josh eclipsed his previous career-high by scoring 22 points on 11-for-13 shooting.

He replaced Jason Collins as the starting center and contributed immediately, putting up a double-double almost every night.

[10] On January 26, after he had played in only two games, it was announced that Boone would miss the rest of the 2012–13 season with a knee injury.

[11] In February 2014, Boone signed with the San Miguel Beermen as an import for the 2014 PBA Commissioner's Cup.

[19] On December 24, 2016, Boone signed with Melbourne United of the Australian National Basketball League as an injury replacement player.

[20] On February 21, 2017, Boone signed with the Hong Kong Eastern Long Lions of the ASEAN Basketball League.

[26] In November 2021, it was announced Boone was on the roster of Canadian club Edmonton Stingers to strengthen their team for their maiden BCL Americas campaign in the 2021–22 season.