Lay a piece of plastic down on the floor.
Flip the chair upside down on the plastic and spray each of its joints, hinges, springs, wheels and moving pieces with an anti-squeak lubricant spray.
Wipe away all dripping lubricant with paper towels and then flip the chair upright. Sit in the chair and move around in it while listening for squeaks. If squeaks persist flip the chair over again and spray any missed areas. Wipe off the excess spray and retest for squeaks. Repeat this process until the squeaks are gone.