You need to pry-off that metal washer on the very end. Use a small flat-headed screwdriver, pop the end into a slot in the washer and gently twist off the washer - careful, it can pop off and go flying around anywhere.
You then need to keep in memory how that joystick metal shaft pulls out of the hole, and watch what direction the plastic tube is facing or it won't fit back on properly.
Once all that is done the large rubber ring will pop out the bottom. You can cover the rubber ring in talcum powder/Baby powder - to stop the squeaking, just be careful of excess powder getting in the joystick itself. You could try a wd40 or similar, not sure it it deteriorates the rubber or not - just remember to wipe off any excess.
One other thing to try is to find a large metal spring, the size of the rubber ring, and use that. The pros of this is that it won't deteriorate or squeak, while the cons are simply finding one.
I have done 3 of my Competition Pro Joysticks now, and they work for a while. The squeak came back on my clear Competition Pros. Haven't tried the metal spring, yet.
