Remoil on the collet? No I don't. I take mine completely apart now and then per the link above. Non seize lube, only on the outside of collet petals. Manual says the inside of the collet is to be kept clean and dry.
Barrel: I use the same method that Yak does. Curleynohair is a mec guru and a heck of a nice guy. You don't want to get any oil or grease on the inside of the collet. If you did use oil it wouldn't last long and your collet would start to stick.
If you do a search on curleynohair,in shotgunworld, he has another thread on cleaning and greasing the shell plate,it made all of my mec's run smoother. I hope this help's ya.. Rick