Hi Pete,
I'll add some of my own thoughts to the mix. Just so you know where I'm coming from, I'm a senior design engineer for a company called Intersil. I'm also a part-time graduate student at Stanford, pursuing an MSEE.
Does a Stanford degree make you more likely to get a position? It depends entirely on the company. There are a few companies that literally will not even interview you unless you've been to an elite school like Stanford, or are the only person on the planet who has some specific skill they need. A few high-profile companies (e.g. Google) have their pick of tens of thousands of applicants, and pretty much only hire other "good ol' boys" from schools like Stanford.
On the other hand, the vast majority of companies will hire you if you seem competent to do the work they need done, regardless of your formal training -- especially after a few years of experience. I know of a number of very competent people who were educated by the Navy or got some kind of vocational training, then worked hard and climbed the ladder of responsibility, and would now be hired by just about any company they wanted to work for. We also have a huge number of people who studied overseas at schools I've never heard of -- some of them might not be highly-esteemed, but the people are still very competent, so it doesn't matter.
In my experience, most employers don't care at all if you got your MSEE from the same school as your BSEE.
Once you get to the higher ranks in major corporations, you'll find a lot of people who have studied all sorts of different fields earlier in their lives. We have people here who did computer security, others who have physics degrees, you name it. It definitely doesn't hurt to have a well-rounded education.
It may even come as a surprise how much of the job interview process is social, even when it's not supposed to be. If the interviewer leaves the room thinking you'd be a fun person to work with, you will likely get an offer. Well-rounded candidates essentially have more to talk about with the interviewer and have a higher likelihood of having something in common with him/her. This makes the interviewer more likely to view you as a peer, and more likely to think favorably of you after he/she leaves the room. I have literally had interviews which included 30 minutes of talking about the kind of telescopes I like to use, and whether or not I like certain kinds of eyepieces -- at an interview for an EE position.
As far as internships are concerned, I see two favorable paths, one of which is a series of summers or semesters interning. The other is to get involved with research projects on your own campus, which you can do year-round. Both give you the same sort of "real world" experience, and both look great on a resume. I was heavily involved in the Hybrid Electric Vehicle Team at Virginia Tech, and the experience was so rewarding that it's still on my resume, six years later. I also wrote several papers on my work for undergraduate research credit, so the project actually helped me graduate earlier.
I also agree that some computer science experience is invaluable for EE work. These days, almost everything is done on a computer, and a good UNIX and scripting skills are just about necessary. I don't intend to push you to do a CS double-major -- that would likely be overkill -- but investing some time in some programming and UNIX classes will definitely pay off in the future.
(Also, I should mention that I was hired for my current position with only a BSEE and about 5 years of experience. I'm attending Stanford remotely, taking my classes online, and my company is paying for it through a tuition-reimbursement program. It'll take me a total of about four years to finish the Master's, but I actually think I'm happier doing it this way, getting paychecks every two weeks, than I would be if I were attending full-time. I also don't think I would have been admitted to Stanford without my experience and corporate backing. There's always another way to get things done!)
Good luck!
- Warren