Speaking on a BBC 6 Music special with Paul Weller, the 32-year-old gold medal-winner revealed that he gave Squire the gift at the band's reunion gig in London in the summer.
A well-known lover of Mod music, Wiggins said: "John's a big cycling fan, he goes out on his bike a lot. He wanted to go out for a ride one day.
"I signed him a yellow jersey from the Tour de France before that gig. I think it made a lot of people happy, them getting back together."
Wiggins became the first British winner of the Tour de France in July this year, and won a gold medal at the London Olympic Games.
The cyclist's amazing year was topped off by winning the BBC Sports Personality of the Year award this month.
Photo gallery - Team GB's London 2012 gold medals:
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