While I don't know anything at all about brine quenching, as an (ex) culinary arts major, I do have $0.02 about boiling eggs!
This is just my theory, completely off the top of my head. When boiling eggs, you have to be careful to not actually boil the water, but keep it just below the steam point, because while pure water can only reach 212f, steam may get much hotter than that, "causing a sulfur reaction in the egg ruining the flavor and showing your incompetence... blah blah blah". Steam would be a terrible medium for quenching, and delaying its formation could allow your steel to quench that much faster (still a guess/theory!). As well, saltwater is a conductor, and perhaps adding the salt allows the quench heat to dissipate over the entire body of water more quickly, rather than stay localized?
Either way, you can tell your friend that the reason he adds a lot of salt to his boiling egg water, is because saltwater is more dense than freshwater. The goal is to make the water so salty, that the eggs float, and never touch the bottom of the pan, which is much hotter "causing a sulfur reaction in the egg ruining the flavor and showing your incompetence..."