Trailing 3-0 early in the second half of the Toolstation Premier Division clash, Barum looked on their way to a crushing defeat.
But Simon Bochenski gave them a glimmer of hope with 25 minutes remaining when he swept a shot between Willand goalkeeper Brian Cann's legs.
Barnstaple still looked to be heading for a third straight defeat, though. That was until James Turner was chopped down in the 90th minute and Squire sent Cann the wrong way from the spot.
And Town's dramatic comeback was complete a minute later when Danny Tapp burst into the area and crossed for Squire to turn the ball in.
Two first-half goals from Alexis Piper had put Willand in a strong position. And when the impressive Danny Baily fired in Rovers' third a minute after half-time, Barnstaple had looked dead and buried.
See Thursday's Journal for a full report.
Barnstaple: Jimmy Thorne, Steve Avery, Steve Shore, Jamie Frickleton (Jake Barwick 46), Danny Tapp, Ian Patchett, Darren Goff (Ben Layland 69), James Turner, Mark Biggs, Kevin Squire, Simon Bochenski (Lewis Irish 83).