Right, I would stick with the E.E. major if that's what you are more interested in. You don't need the CS degree to be a programmer and the E.E. degree would be enough to show that you got the brains.
As for start-up building, all the more reason to give it a go. As with anything, you learn by doing. You know nothing about building your ideas, but once you take a shot you learn a lot more about it. You don't need to shoot for the stars. Try something small.
Maybe I should stick with the E.E. major and work on CompSi&Coding in my free time.