I use a synthetic bicycle chain lube. Attracts less dirt, and is very persistent. Doesn't gum up like "aircraft grade" greases.
The squeeze bottle comes with a thin plastic tube tip. I replace that with a 1/16" tygon tube about 6" long. The trick is to invert the bottle and squeeze about 1/4" of oil into the tube, then turn the bottle upright and propel that one drop of oil with the air compressed in the bottle. That way I can dispense one drop of oil into some of the blind spots without spilling excess oil. A syringe is also a good tool for the job.