The gaiter is the rubber thing you are cleaning, and green oil is good news for you.
As you have lubricating oil "leaking", not power steering fluid, and leaking means there is still some oil in there. What I would do...
Remove the gaiter clips at both ends, turn the gaiter 180 degrees, get an oiling can with oil in it, push the can nozzle under the top of the gaiter at any place you can get to, then squirt oil in, not huge amounts, replace the clips if you can, if not, buy a couple of suitably sized jubilee clips..????