Regina began working professionally on stage in an off-Broadway production of The World of Sholom Aleichem in 1976.

He played Kenickie in a national touring company of the musical Grease, and later appeared briefly in the Broadway production.

He had starring roles in the television series Joe & Valerie from 1978 to 1979 with Char Fontane and Zorro and Son in 1983 with Henry Darrow.

He is perhaps best known for his portrayal of gay sibling Cliff Waters, with Robert Walden and Brandon Maggart as his two brothers, and Philip Charles MacKenzie as his best friend, in Showtime's groundbreaking 1980s sitcom Brothers, which was the first American sitcom with homosexual main characters that dealt with homosexuality in a real way.

He collaborated again with Fred Carpenter and Joanne Tamburro on the story for the 2011 film Jesse, released after his death.