Defender Brindley and striker Gnanduillet both joined the Spireites in January on short-term deals until the end of the season.
Negotiations to extend their stays at the League Two club looked like they might be concluded before the weekend but the players refused to sign the contracts they had been offered, resulting in them being omitted from the squad that travelled to Underhill.
"We tried to secure both players on new contracts last week and were hopeful that the business would be concluded on Friday," Chesterfield chief executive Chris Turner told the club's official website.
"Unfortunately, negotiations broke down when the agency involved advised Richard and Armand against signing the deals they had been offered.
"It was subsequently decided that neither player would travel with the squad to Barnet."
The club added that Brindley and Gnanduillet were to resume training with the rest of the squad on Monday and that both players would be considered for selection for Saturday's home fixture against fellow play-off hopefuls Bradford.