Whenever you hear a squeak and creaks in an upholstery piece (or even a solid wood chair) 95% of the time its caused by wood moving against wood. To eliminate the creak/squeak, you have to tighten the joint. On some pieces, it's relatively easy as you can get access to the joint and run some additional wood screws into it at an angle. Others - such as rungs in a chair - require removing and re-gluing.

Anytime you are adding screws into wood, be SURE to drill a pilot hole prior to running the screw home to avoid splitting the wood. If you do split the wood, the fasten is compromised and not holding, so remove the screw - drill another hole elsewhere and run the screw home again.