Engineering (EECS or MechE): Berkeley, Princeton, Duke, Rice, Brown, or UCLA?

If you want to work in California, UCB or UCLA will be more attractive to smaller employers in California who may not want to travel as far to recruit. If you want to work on Wall Street or in consulting, Princeton is much more highly recruited for that. You will probably find Google, Microsoft, etc. recruiting at all of the schools on your list (and more).

If you are likely to change majors, that increases reasons to favor schools where you can change into your possible new majors more easily. At UCB, if you want to major in CS, you can also do it via L&S CS, but the minimum GPA was recently raised to 3.3 (up from 3.0 which itself was an increase from 2.0 a few years back) due to capacity constraints, so the difference between changing into L&S CS versus EECS is no longer as large as before (though EECS is still competitive after meeting the GPA, while meeting the GPA means that you are in for L&S CS).