Over the course of time the motion of a rocking chair takes its toll on the joints, causing them to come loose and the chair to start squeaking. Other causes usually stem from the floor, especially if it's placed on hardwood. While the easy fix for this problem is to place a rug underneath, fixing the actual rocking chair requires you tighten the joints.
Gently move the rocking chair from side to side without anybody sitting on it. If the frame wobbles, the joints are loose and need to be tightened. If the frame doesn't wobble, the problem could be coming from the rockers at the bottom.
Turn the chair over to reveal the bottom of the rockers. Lay rubber shelf liner over the top and cut it with scissors so it's the same size as the rockers.
Apply a thin layer of carpenter's glue to the bottom of the rockers and attach the rubber shelf lining. Wipe off the excess glue with a rag. Wait for it to dry before you sit on the chair. Drying times vary according to the manufacturer.
Tighten all of the screws holding the rocking chair frame together using a screwdriver. Make sure you don't over-tighten the screws as this could cause the wood to split.
Apply glue to loose joints. Knock the joints out of place using a rubber mallet, apply a thin layer of carpenter's glue to the joint, and put the joint back in place. Clamp the chair together with either straps or bar clamps, and wait for it to dry.
Spray a small amount of WD-40 onto the joints if they are completely secure but still squeak.