Other parts in the musical will also be cast for the Broadway, touring, and international productions.
The trio appeared at the Apple Store in New York's SoHo to perform selections from Broadway's Carole King Musical.
The annual collection period culminated in a December 8 show at the New Amsterdam Theatre.
Broadway creatives take listeners behind the music on their cast albums in a new Ghostlight series.
The hit Broadway musical opens in London in February 2015.
Mueller's Tony-nominated costars Jarrod Spector and Anika Larsen have also extended their contracts.
Campbell will replace Jake Epstein in the role of Gerry Goffin.
The new musical will open in the West End at London's Aldwych Theatre in 2015.
In addition to freebies and prizes, the day will include performances by Joe Iconis & Family, TADA! Theater, and more.