Germann was born in Houston. The family moved to the Lookout Mountain outside of Golden, Colorado. His mother, Marlene Marian (née Faulkner), was a homemaker and his father, Edward A. Germann, was a playwright and professor. While living in Colorado, Germann expressed interest in acting. He starred in stage plays during his middle and high-school years, before moving to New York in 1982.
He started in several Broadway plays while in New York, including a role in 1982 alongside Matthew Broderick in the play Fancy This, which Germann also co-wrote. He received praise for roles in the 1982 musical Chicago, and the 1983 musical The Wizard of Oz. He continued to have roles in plays before seeing a poster for wanted actors for a film in 1985. He moved to Hollywood to find film success, and got his first role in the teen comedy film, The Whoopee Boys in 1986. The supporting role in that film helped Greg's career, so he could star in bigger and mainstream films. His next film was 1990's Child's Play 2, in which he played the character Mattson, in the well-received sequel to one of the most popular horror films of all time. Child's Play 2, and the 1991 romantic comedy Once Around, helped the actor gain mainstream success. In 1990, Germann originated the role of John Hinckley, Jr. in Stephen Sondheim's Assassins.