brainwallet.org is the honey badger of none standard zero outputs...
The problem was that min fee for most clients is 0.0001, so the only way to send 0.0001 is to use a client that lets you make no-fee TXs. 0.0001-0.0001=0
You would have had to spend that 0.0001 with another private key with more BTC on it.
Makes sense.