The band was formed in 1974 and was named after the poorly-received Velvet Underground record. [Squeeze, the VU record was released in 1973 without any of the original members of the band, and was panned by critics and fans alike.] The main creative force behind Squeeze (the band) was the duo of Chris Difford (lyrics) and Glenn Tilbrook (music), who wrote virtually all of the songs, including "Another Nail in My Heart."
"Another Nail in My Heart" is off the 1980 LP Argybargy [Argybargy is British slang for an argument, by the by]. As with most of the early Squeeze singles, it did not break the top 100 in the U.S., although it did reach #17 in the U.K. In what can only be described as a crime against music, none of the early Squeeze singles charted in the U.S. until 1981's "Tempted" (#49). The band would go on to have 2 top 40 hits ("Hourglass," which hit #15 in 1987 and "853-5937," which peaked at #32 the following year). A less polite person that I might suggest that the relative success of these later songs could be considered signs of The Universe having a sense of humor ... or irony.
In any event, the band broke up in 1982 (although Difford and Tilbrook continued to work together and even released an album in 1984). Squeeze subsequently reformed in 1985 and continued working together until 1999, then broke up (again) before reforming (again) in 2007. I believe they are together as of this writing.
The video of "Another Nail in My Heart" is a solid early video with the band performing (and hanging out at the pub) while keyboardist Jools Holland pushes a piano around London. I particularly like the way that they shot the guitar solo (starting around 0:52), which is an absolutely superb piece of musicianship.
For folks who are interested in learning more about the band, their greatest hits record, Singles – 45's and Under is highly recommended.