The 28-year-old, who had a three-month loan spell with The Shakers last season, has signed a one-year contract with the Gigg Lane club.
Challinor cancelled the remaining 12 months of his contract with The Hatters on Tuesday before sealing his move to Bury.
The contract has the option for an extension for a further year and Barrow was delighted to land Challinor ahead of next season.
"You come across these sorts of players as a manager," Barrow told the club's official website.
"I had Colin Greenhall at Chester and he was good for them.
"They are diamonds to a manager, especially when they help you on and off the field like Chally has done.
"It's not just his playing ability, but everything else he offers to the club."
