Quinn is linked with Nantwich through their manager, Steve Davis, whom Quinn managed when at neighbours Northwich.
The Nantwich boss said: "I am delighted with the signing, he will be a great asset to the team."
Nantwich play in the North West Counties Division One, which is the fifth tier below the League Two. They are currently languishing in 15th position with only four wins from 15 games.
Quinn has a good goalscoring pedigree, having played 46 times for Northern Ireland. He turned out for the Shrews last season in the Conference and bagged six goals from eleven starts.
Quinn had been linked with a return to coaching at Peterborough, with boss Barry Fry keen to make use of his availability. Fry said: "He'd love to come here and I'd love to have him but the finances here at the moment would make it difficult."
There remains the possibility of Quinn combining the two roles.