The trick to silver soldering stainless steel is to use Harris Stay-Clean Liquid Flux part number SCLF4, EDP (electronic data processing?) number of 40002. The UPC (bar code) is 6 84032 00013 1. This is their most active soldering flux formulation. You may be disappointed in the performance of a less concentrated flux. I buy this flux at the local welding shop but it is also available on industrial supply websites and on Amazon as: Harris SCLF4 Stay Clean Soldering Flux, 4 oz.
The solder I use is either Alph Fry 62964 4% silver solder or even better is Kester SN95AG05 #66/44 which is 5% silver. If you don't need 1 pound of the 5% solder you can buy small quantities of it on eBay.
You can get detailed technical information and an MSDS for the flux from:
http://www.harrisproductsgroup.com/en/Products/Alloys/Soldering/Fluxes/Stay-Clean-Liquid-Flux.aspx
(or
http://www.harrisproductsgroup.com > alloys > Soldering > Fluxes > Stay-Clean® Liquid Flux)
This flux contains Muriatic Acid which is another name for Hydrochloric acid or hydrogen chloride (HCl) which will produce free chlorine gas when heated.
Wear a respirator mask rated for hydrogen chloride (Hydrochloric acid / HCl) vapor and chlorine gas. The average paint spray mask is probably not rated for these vapors. The 3M 8246 Particulate Respirator available on Amazon and eBay in single quantity is rated for nuisance level acid vapors
http://solutions.3m.com > safety and security > personal protective equipment > disposable respiratory protection (picture) > Nuisance Level Acid Gas
Flux residue is corrosive and a post-solder flux removal procedure should be employed. Because of this it is not recommended for use in electrical or electronic applications.
FLUX RESIDUE REMOVAL
Step #1
Thoroughly soak and wash soldered parts in a solution made up of one to two ounces Tri-sodium Phosphate (TSP), Spic and
Span (powdered) or Bicarbonate of soda added to one gallon of water heated to 120° F (49° C) to neutralize the acid residue. Rinse these alkaline cleaning solutions off thoroughly because they are almost as corrosive as the hydrochloric acid.
I keep my bottle of this stuff in an old soup can that gets seriously rusted from the vapor of the flux.