I think the employers you mentioned (GE, Lockheed, etc) will be more interested in the specific expertise you develop in your PhD research, than in the department you graduate from. If Lockheed is looking for a PhD to take a leadership role in radar devices, for example, they'll look for someone who has a track record of generating strong, original and relevant research in that specific field. Whether they took the core EE courses or the core physics courses isn't of primary concern, IMO.
That being said, engineering departments (in my admittedly limited experience) generally have much closer ties to industry. Some of my engineering PhD friends are conducting their research half at a university, and half in an R&D department at companies like Boeing, Lockheed, GE, Schlumberger, BP, etc. Those types of connections definitely open doors for future job opportunities that PhD students who work solely in academia miss out on (and make finding funding easier, too).
Anyway, I think grae's idea of talking to people who hire PhDs at the big corporations you're interested in working for is a good one. I'd also suggest applying to a range of departments, from EE to applied physics to physics. See where you get accepted, and talk to potential advisors. Get an idea for the type of research you could do with them (this depends highly on the professor's interests and funding sources) and the jobs their students take after graduating. It's good to have clear goals at this point, but you don't have to figure it out exactly yet.