Similar to playing "chicken" with an opposing car on a road, "poop chicken" begins when you and someone else walk into a public restroom at the same time, both with the intention to take a shit. You go into your separate stalls and then sit there, waiting for the other to leave so that you can do your business in peace. The first person to just get up and leave without pooping loses the game of poop chicken.
Oh man. I had to take a shit so bad today at work, but some woman from another office on our floor walked into the bathroom at the same time, so we ended up just playing poop chicken for a few minutes until she finally gave up and left. I was victorious and then proceeded to shit in peace.