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... people really need to get some reading comprehension. Nowhere does the article Tracy linked say that any university is eliminating your usual multiple-stall, conventionally gendered men's and women's rooms, or not building them in new buildings.
What's happening is pretty simple: older single-occupancy bathrooms that already exist, where only one person can be inside at a time and you lock the door, are having unnecessary "men" and "women" signs replaced with a sign that says "bathroom." Anyone who's not incredibly uptight knows that these kinds of bathrooms are already used, and have been for years, by both men and women, because it means you don't have to wait in line and for god's sake, it's not like anyone else is going to be in there anyway. Ironically, these sorts of bathrooms are the safest for those who don't want to encounter transgender people, your personal stalker of either gender, or Larry Craig while in the bathroom. It's really better for everyone, especially when you consider that such bathrooms can serve as easily accessible handicapped facilities as well as changing-rooms that can be accessed by moms OR dads.
The Dulles International Airport in Washington, DC has a really good example of a gender-neutral bathroom that clearly advertises all of the above -- family changing station, handicapped accessible, single-occupancy bathroom with no "men" or "women sign." It's right next to your usual multiple-occupancy men's and women's rooms. That's hopefully the kind of setup we'll see in more places.
Plus, from what I understand advocates at universities are not insisting anything close to "all bathrooms must be gender-neutral." It's something more like, one per building, and letting people know where it is if they want to use it. That's not much to ask for, but of course some right-wingers spin it into an approval of transgender people (horrors!)