I don't think you have to be a woman to get it. My male DP read the book many years ago, before I met him, and is enjoying the TV show as much as me (if that's the right word).
I think it's less to do with gender than with being ill-informed or a bit hard of thinking; as PPs have pointed out, it's not 'just' about women's rights –it could apply, and does, to anyone's civil rights and the consequences of the loss of them.
I thought June's friend in the flashback was a bit hard on Luke; 'I'll take care of you' is a totally normal thing to say to someone you love. I'D probably say that to my DP in that situation were the roles reversed. And then she growled at him for referring to June as 'my wife' –FFS, that's just how people refer to their spouses. That has been the only thing that felt heavy-handed to me.
For my money the scene where she tries to pay for coffee was the most powerful yet. I could easily imagine the guy behind the counter yelling, rather than 'Fucking sluts', 'Fucking P*kis' or 'Fucking fags' or something. Societies/pockets of society where it is 'OK' to hurl abuse at people over their identities are very much alive and kicking; all it takes is for those in authority, and the society at large, to set up and allow the conditions for those attitudes to flourish.