Very well said cailindana.
Of course females need safe female-only spaces.
Personally, I do not see a problem with a place that only cuts men's hair and men's beards saying that they do not want females in their establishment. They may lose trade but so be it. They may also pick up trade. If a hairdresser wants to only cut women's hair and wants to be women only, again, so be it. Lots of spas and the like could be male or female only. To me that seems fine.
I think sending out a pregnant women seems a bit much, but imagine if there was a female only space and a man on crutches wanted to sit there or whatever. I would ideally not want the females in that space to be compromised by having to have a male in that space. Being pregnant is not being ill. Presumably, being pregnant doesn't mean you can't walk or drive to a local cafe or go home.
Male only spaces are not for the same reasons as female only spaces. But males may like the chance to be together just as females want the chance to be together. it's like Guides/Brownies/Girls' Brigade. etc, I want those to be for females only. So I don't have a problem with Boys' Brigade.
I certainly would not want there to be a whole lot of places in society that are for males or females only, I would rather most places were for everyone. But some places that segregate on sex, seems fair and not unreasonable.