Egg white is a component in some "ground" recipes used in finishing stringed instruments, and it is sometimes used to fill the grain on guitars etc. It also get some use in marquetry, because it will keep colors in adjacent inlays from bleeding. You might read up on "vernis bianca"--a luthier's ground. I've never tried it on a rifle.
If you are going to use it as a sealer, you might try the approach Cennini recommended for "varnishing" carvings (Cennini was a 15th century Italian, but his book was reprinted by Dover as The Craftsman's Handbook). Whip the egg whites into a hard foam, then let them sit over night. The next day, apply the "distillate" (the watery liquid in the bottom of your bowl) to the carving with a soft brush and let dry.
I've experimented with this a bit on practice carvings, haven't actually tried it on a rifle (and wouldn't without further experimentation). I'd say definitely stain the rifle first, after all--the egg white is being used as a sealer.
Good luck!