At the moment Lightning can't do external callouts. It can call an Apex method that does the callout, but it has to be a synchronous callout because there's no view state. And there's a limit of only a certain number (20?) of simultaneous synchronous callouts from Apex to the same host.
One dev at my last company had a clever workaround. He put in a hidden iFrame containing a VisualForce page, and designed it such that it could pass messages back and forth with the Lightning component that was hosting it. So Lightning told VisualForce to make the asynchronous callout and got the message back.
Some thoughts from here: