I heard from Adafruit that "Raspbian Wheezy does not have the updated firmware for this much RAM" in regards to the Pi Zero. This doesn't make any sense to me, seeing as I run Wheezy on my Pi 2, which has more RAM. They also said that Raspbian Jessie is the only Raspbian compatible with the Pi Zero. Is this true?
The last official Raspbian Wheezy image was 2015-05-07 and official images moved to Jessie afterwards. This was 5 months before the Zero was released, so didn't contain the firmware for the Zero or the pi-3 for that matter.
This can be fixed by booting them in an older pi, then running
sudo apt-get update && sudo apt-get upgrade
The SD card will then boot fine in a Pi Zero / Pi 3
I did exactly that and have uploaded the image here for Pi Zero and pi3 users: