Petroleum oils were largely unknown during the mountain man era, so "gun oil" would have been some sort of animal or vegetable oil, probably sperm whale oil by preference. Bear oil would be my guess as a preferred substitute, but whatever kind of fat one gets from a beaver or tallow from deer and bison are also possibilities.
No idea how they carried it, but some sort of horn container sounds plausible. If tallow, the box of the rifle is a possibility too.
I've used lard as a preservative grease, and while I want to do some more experimenting, so far I've found it to be a perfectly adequate grease for preserving both metal and wood. Doesn't smell bad, either...Veggie oil gets sticky pretty fast, though. Not sure I care for it, and given the complaints I've heard about both "bore butter" and "frog lube" (both of which I think are based on vegetable oils of some sort) I'd stay away from them as a lubricant. I've been told that animal fats will rot wood, but I haven't seen that so far.