The only documented widespread use urine in cleaning gun barrels is from the Russians in WW2 and it was more for the conservation of precious water supplies and fuel (to melt ice/snow) than it was for the fact it was an effective cleaner.
Also, urine in past times was collected and used in place of what we would call "modern chemicals" for cleaning and etching certain products as well as other industrial endeavors, so the fact it may have found its way to cleaning a few gun barrels is not really that much of a stretch historically. Could very well even been used in the production of gun barrels as one thing it was used for was cleaning impurities off of steel during its production.
As far as the salts in urine, and adding another step to flush them out. That's easy, warm water works better than cold, and when you are unable or unwilling to start a fire to heat water, you can use urine, which is considerably warmer than ambient temperatures, especially in winter, for the heavy cleaning and then follow with a quick rinse of cold water. Thus saving fuel, keeping your position hidden, etc.
Hot water makes much more sense these days though.