Reading the American media, I haven't seen any connection between this trend and vegetarianism except the name, similar stereotypes of "herbivores" and vegetarians (that they're weak, effete, and unmasculine,) and a comment on that page that says there's currently a fad for vegetarianism in Japan. I read a different and more likely-sounding explanation for the name, which is that in Japanese a sexual relationship can be called a "friendship of the flesh."