Tom,
As Ellis told you, Some types of stainless can be soldered with some types of acid flux and silver solder, but there are alot of differences in the types of steel used in guns. I have sucessfully soldered on a stainless barrel using regular Brownell Hi Force 44 and their standard Comet flux, but I don't know what the make up of that stainless steel barrel was, (I got it by mistake). If you try it, be sure you tin everything with solder first.
If you are going to do much soldering on guns you need to get some of the standard Hi Force 44 and try it on your barrel steel just to see if it will do what you want. You definately want to get the barrels washed off immediately after soldering. I rinse them in Hot water from the tap first,(we have our water heater turned WAY up), then I boil them in a stainless steel tank I had welded up. It is just long enough to stretch across two burners of our stove. I add very hot water to the tank and a copious amount of baking soda.
I boil the barrel or barrels for 30 minuites at a gentle rolling boil. Then I pull them out and run more hot water rinse over them and dry them immediately with a electric hair dryer set on High/Hot. That drys them off very quickly. If I am bluing them I do that job next. Last, you need to get some good oil on them ASAP to stop after rusting. I check and re-oil them every few days just to make sure no rust is forming. I do that for a month or so.
If your barrel is truely stainless, you may not be able to come up with anything that will blue it. I know that there are commercial bluing products out there that will blacken or blue a stainless barrel, but I have never been able to find any of it to buy. Again, that product may or may not work on your stainless barrel, depending on it's composition. Good Luck to you on your project.