Basically, "bluing", "browning" and "blackening" are simply refined rust, and the whole intent of stainless steel is that it is highly rust resistant. I have seen several products that do work to chemically blacken stainless steel (not paints) but the gunshop products I have tried don't work any better than cold blue works on regular steel. The darkening lasts for a while but will rub right off in a short time.
Factories do blue stainless steel barrels, but seem to keep the process a closely guarded secret. The problem has been gotten around in other ways, among them anodizing (yes, it works with steel as well as with aluminum), and plating the steel with iron, then bluing the iron. Neither is really a DIY type of operation.
Jim