Yes both should go in. I am not exactly sure as to what would happen (Maybe damage or something to the card) but it would be best to have both in. The 560 should probably have two 6 pin pcie slots(If your power supply does not have these then either the card came with molex to 6 pin wires or you could buy them for cheap). Without plugging in both cables, the card will not work fully, or it may not work at all.
the 560 needs both slots. gpus need the 75w pci-e cord in order to power the card to full potentials(which is the one thing i think nvidia needs another of as the strongest single 6 pin connector gpu i can think of for nvidia is the 550ti, wheras AMD has the 7850, which are on vastly different performance levels)