Rifle, it's funny that you should mention urine browning. I actually "browned" a stainless steel AMT .380 Backup that way once. The little gun used to eject cases back into my face. In one magazine full, it landed one in my shirt pocket, on top of my shooting glasses, and lodged one on top of my ear. After a burned nipple, singed eyebrow and reddened ear, I threw it down in disgust, and well... A few weeks later, I got to feeling guilty knowing that if someone discovered my desert shooting spot, they might find the little gun and do who knows what with it, so I went to rescue it. After cleaning and oiling, it has taken on a nice, permanent brown. So, it works even for stainless.
I can almost see how browning was invented. A mountain man takes a shot at a charging bear with his flinter. It misfires! After getting stomped and mauled, the mountain man throws down his rifle in disgust and hoses it down and heads off to heal up. Next season he stumbles across it, cleans it up and voila.
On a serious note, I'd highly suggest against heating the barrel red hot and quenching. That hardens the steel and can cause it to become brittle. Softer metals are more resilient and less likely to shatter under the stress of firing. When a soft barrel lets go, it splits. When a hardened one lets go, it fragments. The barrel makers use softer steel for a reason.