I hunt in WAY sub zero temps sometimes, the only way the gun will function 110% at 15-20 below zero is DRY, BONE DRY. Now, if temps will not get below about 15* what Dave said is the best way to do it. Let it hang for a day or two if you can. Otherwise just shoot some lube in it and go.
In preperation for a super cold weather hunt the gunscrubber type sprays are what I use to get all the lube out, they work well.
Don't worry about rust or wear when hunting with it dry, in that kind of cold everything is frozen and the number of shots taken when hunting are not all that great. So long as you don't warm the shotgun in the house or car rust will not start. If you take it in the house or warm it in the car you had better get the lube and rust protection on it ASAP as the entire gun will be wet from condensation.