A squeaky rocking chair is often the result of loose parts. Wood squeaks when rubbed under pressure, such as when someone sits in the chair and begins to rock. You may notice the chair makes no sound when stationary, even if someone is sitting in it. But when the rocking begins, so does that hair-raising squeak.
Isolate the squeak if possible. Have someone sit in the chair and rock while you pinpoint the source of the trouble. This will often be a wooden rod at the point where the rod inserts into the rocker on each side of the chair, or the crossbars parallel to the seat. Wooden rods fit into holes on the top of the curved rockers that sit on the floor. It is at this joint between the rod and the rocker where the squeaking often occurs.
Crossbars are also rods. These provide stability for the rocker by fitting into the vertical rods. Squeaking can occur at these joints as well.
Turn the chair over and check each dowel for looseness at the joint where it fits into a rocker or the seat.
Apply a short burst of cooking oil spray to the squeaking joint and test the chair to see whether this solves the problem.
Pour a half tsp of Murphy Oil Soap onto a towel and rub the soap into the squeaking wood if Step 3 doesn't fix the problem.
Spray short, controlled bursts of WD-40 onto metal parts, such as the steel hinges on glider rockers, to stop squeaking.
Apply expanding wood glue into the joints of loose wooden parts. The glue expands as it dries to form a seal and prevent wooden rocker parts from moving, which should put an end to that squeaking. The glue comes in a syringe bottle or similar dispenser and is available at hardware and home improvement stores.