The Velvet Underground — which at that time consisted of Doug Yule (vocals/guitar), Willie Alexander (keyboards/vocals), Walter Powers (bass guitar) and Maureen Tucker (drums) — had toured the United Kingdom and The Netherlands in October and November of 1971 to support its latest album, Loaded, which had seen a European release in March of that year. The plan was to record a second and final album for their record company, Atlantic Records, afterwards,  but Atlantic had lost faith and decided to issue an archive recording from 1970 featuring Lou Reed, Live at Max's Kansas City, instead. 
After the tour, band manager Steve Sesnick managed to get a recording deal with Polydor UK to record a final Velvet Underground album. Alexander, Powers and Tucker were sent back to the United States by Sesnick, however, presumably to retain maximum control over the finished product.  Thus, Yule was the only Velvet to actually perform on the album.
The eleven songs that make up Squeeze were recorded with the help of Deep Purple drummer Ian Paice, with additional occasional saxophone and backing singer assistance. They range from Beatles-like whimsy ("Crash") via pop to typical 70s rock ("Mean Old Man").
Squeeze was recorded in the autumn of 1972 and released in the United Kingdom, France, Germany (all 1973) and Spain (1974). No singles were taken off it and the album did not chart. The album saw a number of re-issues in France during the 1970s and early 1980s. It has been out of print since, and has not been issued on compact disc.