My one chair that squeaked was fixed by using a welder to repair the cracked steel plates of the base, and then painting and oiling the rest of it. Seeing as it is around 40 years old, having fatigue was not exactly unexpected, as I got it on an auction a long time ago. The other I ust put a drop of oil and some grease on the swivel point every so often, as the original grease they used is both mostly turned to powder, and also was there in such a minimal amount that 1 small 50g tube would likely have done 500 chairs. I used some nice stringy axle grease blended with black moly loaded CV joint grease, as it will stick to anything ( including PTFE) and is a really good low speed lubricant, though it is stringy and tacky forever. Would use fork lift chain grease but have none to hand, that stuff really sticks, but as it quickly gets a dust coat it is not sticky in use.
With plastic parts dry chain lube for bicycles works well, it both is clear and generally does not harm plastic, though you will have to be careful in application, as it is like glue in it's ability to attach to surfaces.