He is best known for starring as Billy Bigelow in the third Broadway revival of Carousel and as Haywood Patterson in Kander and Ebb's The Scottsboro Boys, both of which earned him Tony Award nominations.

[10] Henry made his acting debut when he co-starred as Judas/John the Baptist in the Paper Mill Playhouse (Milburn, New Jersey) production of Godspell in 2006.

From March 24, 2010 to July 20, 2010 he played "Favorite Son" in the original Broadway cast of the Green Day musical American Idiot.

[19][20] Henry then joined the Broadway cast of The Scottsboro Boys, directed and choreographed by Susan Stroman with a score by John Kander and Fred Ebb.

[22] The Star Tribune wrote of his performance, "Joshua Henry ... ignites the role of Haywood Patterson as the lightning rod for our identification with the men.

"[23] Charles Isherwood in The New York Times asserted that Henry "gives a performance of keen intensity as Haywood Patterson, an illiterate young man who becomes the focus of our sympathy as he steadfastly refuses to sign his name to a lie in order to obtain parole.

Mr. Henry performs Haywood’s ballad of hopeless yearning, "Go Back Home," with a powerful simplicity that slashes through the evening's artifice.

[32] Henry performed the role of Jake in the American Repertory Theater's production of Porgy and Bess, which began previews August 17, 2011, at the Loeb Drama Center in Cambridge, Massachusetts.

Henry starred as Jamie alongside Cynthia Erivo in a one-night benefit concert performance of Jason Robert Brown's The Last Five Years on September 12, 2016.

[36] Henry originated the roles of Hercules Mulligan, James Madison, and King George III in the 2013 Vassar College workshop of Lin-Manuel Miranda’s Hamilton.

[37] In 2016 and 2017, Henry starred in the lead role of Aaron Burr in the Chicago production of Hamilton: An American Musical at the CIBC Theatre.

[4] On November 18, 2020, it was announced Henry would reprise his role of Burr in a special performance with the cast of Hamilton for the Macy's Thanksgiving Day Parade.

[38] In 2018, Henry starred and gained wide-spread acclaim as Billy Bigelow alongside Jessie Mueller in the third Broadway revival of Rodgers and Hammerstein's Carousel, for which he received a Tony Award nomination.

During the run he would star opposite Gavin Creel, Cheyenne Jackson, Karl, Sara Bareilles, Brian d'Arcy James, Patina Miller, Stephanie J.

[46] In May 2024, it was announced Henry would star as Coalhouse Walker Jr. in a production of Ragtime at New York City Center opposite Kalukango, Caissie Levy and Brandon Uranowitz.

He starred alongside H.E.R., Josh Groban, Martin Short, David Alan Grier, Shania Twain, Rizwan Manji, and Rita Moreno.

[52] On January 16, 2011, Henry, Laura Osnes, and Natalie Weiss were featured performers in Kait Kerrigan and Bree Lowdermilk's You Made This Tour concert at the Laurie Beechman Theatre.