The Blue Jackets also assigned forward Michael Chaput to the Falcons on Wednesday.
McKenna will back up Curtis McElhinney, who will start in place of the injured Sergei Bobrovsky. Bobrovsky was hurt in the third period of Tuesday night's 1-0 victory over Tampa Bay. He has an unspecified lower-body injury.
McKenna has appeared in 17 career NHL games with Tampa Bay and New Jersey, with a record of 4-9-1 to go along with a 3.49 goals against average and .887 save-percentage.
Chaput had an assist and two penalty minutes in 17 games with the Blue Jackets.