I would say it partly depends on how you want be known at the school where you are applying. Fill it out the first way, and you'll be known as Juan González at that school. That's how your name will likely be printed on class rosters, and that's what professors will see when you've registered for their class. If you'd rather be known as Juan Eduardo, it'll be a constant uphill battle explaining that to your instructors each semester.
However, if you put Juan Eduardo as your first name, most professors will see that in the first name field, and they'll be more likely call you Juan Eduardo on the first day of class.
As for your last name, hyphenated last names are not uncommon in the U.S. Therefore, if you are applying to a U.S. school, it might be worth hyphenating the last name – but not if you feel like you are compromising your own identity to do so. If you feel like your last name is González Rodríguez (no hyphen), then list it that way. People might get confused initially, but they will adapt. Moreover, if they've never seen a 2+2 name before, some might even be thankful to learn something new.
Part of this is a tradeoff between how much you want to retain your name as it is, and how much you want to fit your name into a culture that is more accustomed to First MI Last.
As a footnote, perhaps a university registrar will read this question, and begin to wonder if they should tinker with their institution's online application form, to be more accomodating to people from other cultures.