My advise has changed since last time I posted here: don't waste your time on a dual degree/major.
Pick one thing that is dear the most to you and go through bachelors and masters and phd with it. A PhD in any field will give you a flexibility to apply yourself to vast number of similar fields, and I would recommend that.
A Physics major over EE major will have intro courses from all engineering disciplines and might also include a few grad courses in theoretical areas that arent in any engineering degrees. On the other hand, an engineering degree is a professional degree that will allow you to have a lucrative starting salary as well as other benefits that are mostly applicable to engineers. Sure you can get into engineering with a BS in Physics, but starting salary won't be that of an engineer with a BS. You might start at 30-35k/year and work for a few years before you'll be promoted to an engineer - and even then you can't practice without a PE license.
So pick which one is really important to you, and stick with it. Get the basics out of the way as soon as possible and go for advanced stuff