Find the part of the spring that is making the noise. Sit on the chair and pay attention to where the noise originates. It will be easier to have someone else sit in the chair and rock while you stoop to listen for the squeaking area.
Turn the chair upside down on the floor so that you can easily access the bottom spring area.
Apply a lubricating spray to the exact area on the spring where you notice the squeak. In a pinch, you can also use kitchen cooking spray oil. Rub the oil over all the bolts and moving parts.
Rub the squeaky joints and springs with petroleum jelly or mineral oil if the lubricating oil does not take the squeak away.
Heat the areas you have rubbed with mineral oil or petroleum jelly with a hair dryer to warm the substance so it runs down into the springs to reduce squeaking.
Melt candle wax and rub it into the wooden area around the springs with your fingers. Sit on the chair and immediately start rocking to help the wax sink into the wood. If you are uncomfortable using hot wax, you could also use furniture polish or a bar of soap to lubricate the wood.
Lubricate your springs every month or so to prevent squeaking.