LOL. You'd be making more of a protest/statement by just quitting your job. If you're against tolerance and respect for other human beings, go work for Walmart. Oh wait, I forgot, Walmart has the same bathroom policy too, as do most public businesses. Damn, seems like you just can't get away from those pesky transgendered people (people who are obviously all terrible, disturbed perverted monsters in your opinion). In public bathrooms. Emphasis on public, because you know, they're meant for the public to use, and is plain outright discrimination to say that a group of people don't have the right to use a public bathroom.
By signing that petition, here's what you're actually saying. "I am a closed minded, intolerant, hateful, naive bigot against transgendered people because all transgendered people are perverted, disturbed, monsters. Transgendered people should not have the same rights as me, and therefore transgendered people are lesser than me and their life and their rights should not be respected as much as my own are."