I'd expect feminism and female equality to match racial equality of the 70s as well, more or less.
If not more so--in OTL, federal "liberalizing" efforts re: race weren't especially effective in the South, but I don't know of any problems re: sex.
(In fact, in "The Good War," I remember reading about hillbilly women working in a munitions plant. No mention of outraged redneck men pitching fits their womenfolk had independent incomes.)
I wonder if in TTL the pro-choice movement will be able to tar the pro-life movement with the Nazis' extreme-natalist attitudes? Of course, the pro-lifers can hit back over eugenics. Both sides might invoke the Nazis against the other.
Pro-Choice: If you don't support legalization of abortion, you want women to be used as breeders just like the Nazis did!
Pro-Life: By supporting aborting children who are "unfit" (genetic diseases or fetal injury) or whose fathers are bad (rape or abandonment), you're eugenicist just like the Nazis!
(Note: This is not to denounce either position as being deliberately evil. Pro-lifers aren't Taliban and pro-choicers aren't eugenicists. Nobody try to turn this into an abortion debate.)