The founding members were Chris Difford (guitar, vocals), Glenn Tilbrook (vocals, guitar), Jools Holland (piano), and Paul Gunn (drums). Gilson Lavis replaced Gunn on drums and Harry Kakoulli joined on bass in 1976. Their debut album was produced by John Cale for A&M Records.
John Bentley replaced Harry Kakoulli on bass after the latter's firing in 1978. Jools Holland was next to exit, being replaced with Paul Carrack in 1980. East Side Story, perhaps Squeeze's most successful album, was released in 1981, produced by Elvis Costello and Roger Bechirian, and featured Carrack's lead vocals on "Tempted." However, Carrack would leave after the release of East Side Story, and was replaced on keyboards by John Snow. After Sweets from a Stranger was released the next year, Difford and Tilbrook broke up the band.
However, 1985 saw the return of Squeeze, now featuring Difford, Tilbrook, Holland, Lavis, and Kevin Wilkinson on bass. Andy Metcalfe of the Soft Boys and Robyn Hitchcock and the Egyptians joined the next year to supplement Holland's keyboards. His tenure with the band would only last a year, however.
Jools Holland left Squeeze again in 1989, and Gilson Lavis left in 1992. By this point Squeeze was basically a trade name for Difford and Tilbrook plus sidemen. Players with Squeeze during the 1990s included drummer Pete Thomas and pianist Steve Nieve of Elvis Costello's Attractions, drummer Kevin Wilkinson, Don Snow, Hilaire Penda, Ashley Soan, and Jools's brother Christopher Holland. Chris Difford effectively broke up the band in 1999 with his announcement that he was taking a hiatus from working with Glenn.