There will ALWAYS be demand for GOOD engineers. If you're willing to invest time and effort and pursue a degree from a good school, I'm sure you could find work later on. And as one poster here wisely said, it's only the initial phase that should trouble you, when you're experienced it's much easier to come up with a job.
I wouldn't worry too much about outsourcing. You'll see the trend stablizing. Outsourcing usually does not yield innovation, it's good for dirty works you wouldn't wanna do anyway but when it comes to stuff important to the companies I'm sure it would remain on US soil.
Also, I'd believe engineering has a much potential income, and great places to work at. High tech workers get alot of perks, nice offices, etc. Also, what would be your career path as a nurse compared to that of an engineer?