Cannot login to OS X 10.9 because password field disappears after clicking username
That is odd and I have not seen it before, the only thing I can think of is to boot in Safe Mode in the hopes that there is something running on your Mac that is causing that or interfering with login.

Boot your Mac holding Shift untill the apple logo pops up then release the key and let it boot. It will likely take longer to boot and eventually you will see "Safe Mode" on the boot/login screen. Try it then.