It would be a nice world if such a place as a gender-neutral bathroom was commonplace but it is unlikely as long as men act as men and women act accordingly. A most unlikely event though, now that I live as a woman, I would be very uncomfortable if a man was in the next cubicle to me or weeing into a urinal with his willy exposed as I exit my booth after doing my stuff.
As a matter of record, I must qualify my comments by saying that I use unisex facilities due to my disabilities and there have been no problems owing to the fact that it is a one person facility.