If you are interested in Engineering at private schools that meet full need (and you have a lot of need), a more sensible list is something like this:
Princeton (it is the one among HYP with what is typically considered a strong Engineering prog)
Stanford
MIT
Caltech
Cornell
Carnegie-Mellon
Rice
Amherst College (they meet full need for internationals, and you can do a 5-year BSE with Dartmouth. It likely is an easier admit than Yale or Harvard, though – like H and Y – it is not particularly well known for Engineering.)
Check those schools to see how they do with financial aid for internationals.