Honestly, if you can get an engineering degree from a top school like MIT or something, you will be able to dictate where your career goes. You'll be able to pick and choose, and sometimes create that perfect job. And even if you can get an engineering degree from a decent school, you'll get a job immesiately; possibly before you graduate if you work a good internship.
My dad got a degree in petroleum engineering, but the company had amazing support, and ended up paying for an MBA and then a law degree, so my dad transitioned around as he wanted. One thing he has mentionedone, though, is that he would not recommend someone go into that industry now, because, unless you went to Harvard, you're not working for ExxonMobil or anything. That industry has shrunk since he got into it in the late 70s.
My dad, brother, and sister in law all have law degrees, and it has made very clear that work experience makes the biggest difference with respect to pay etc. However, where you got your degree is so important, so find where you are far above average and hammer on that. Much farther down, in third place, would be internships and the experience and connections you make there. If you can land a killer internship the summer before you graduate you may be employed in advance of graduation, which is sweet if you are on school loans etc.