Regardless of what ojbections neo-cons may have, unisex bathroom are likely here to stay. You might as well get used to it. Despite all of these ground rules, a bathroom is a bathroom. Men using a bathroom isn't always a pleasant thing: if you've ever used the toilet at a Chipotle, Five Guys, or a state fair (shudder), then you know what we're talking about.
It is not only okay, it is your duty to use the bathroom as if it were any other bathroom. Proudly march into the facilities and utilize them to your heart's content. Don't just hold it until you get back to your place out of a sense of courtesy. Do your part for the advancement of civilization and proudly where men and women have not both gone at the same time before.