Naftush:
“Please show me a requirement in halacha, or in plain logic, to police, segregate, and intimidate young people to the extent of forbidding the exchange of the words “gut Shabbos.””
With pleasure. Shulchan Oruch, Even Ha’ezer, siman chuf aleph, sif katan alef: “Tzurich adam lisrachaik mei’hanushim moed moed, a man must distance himself from women very, very much.” Sif katan vav: “Ain shoelin bi’ shalom eisha klal”, which, simply read, means “do not greet a woman at all”.
Sam2 & oomis1105:
“Hevei makdim shalom li’kol adam” could easily be translated as “greet every man”, not “every person”. (The Mishna is obviously addressed only to men; however, one would imagine that the same would apply to a woman greeting a woman.)