Aeronautical Engineering (or Aerospace Engineering) are specialized subsets of mechanical engineering.
The problem with those fields is that they are linked to aviation, which experiences lots of up-and-down business cycles. If you go to work as an aerospace engineer you can probably expect to be laid off more than once in your career. So being in aerospace engineering isn't really much of a "backup" to being a pilot.
The year I grauated from college with a mechanical engineering degree, my school had about 30 aerospace engineering graduates, and only about 4 of them got degrees in that field right away.
Unless you really want to work specifically on aircraft design, e.g. aerodynamics or aircraft structures, you do not have to major in aeronautical or aerospace engineering. Aerospace companines hire plenty of mechanical and electrical engineers.
Let me begin by saying that I don't know what this person was thinking when giving you such bad advice. I hate ignorance above all.
I have a degree in Aerospace Engineering and a passion for flying and have recently gotten hired by a regional. I was previously a Satellite engineer working on pretty cool things. Here are the fact, the Aerospace industry is facing much worse shortages for AE's than aviation is facing for pilots, therefore it is a great backup as a career to flying. Why, because all of the baby boomer AE's are also retiring and there has not been enough people moving up the ranks to take over, plus look at the growth in the industry in general. I know this because I worked in the industry and did the re-search.
Also, no one says you have to be in the aviation or aerospace industry to utilize your degree. You will learn about Aerodynamics, fluids dynamics. that can be used in wind tunnel testing for cars manufacturers, street vehicles or more exciting F-1, indy and nascar. How about becoming a test pilot once you get some years of flying experience.
Other venues are roller coasters, ships, buildings, there are a lot of venues you can take. You like law? Get your law degree and now you can do patent law as an AE, making more than you'll ever make as a pilot, 80 hr a week too, but that's another story.
Yes, we do take ME's and working in the satellite world where there were other ME's. I can tell you that they couldn't begin to grasp orbital mechanics. The reason they were hired because there was a shortage of AE applicants.
Not a bash on ME's because there are many career fields that they are better suited for than an AE. I personally think that as an AE you will get a little of everything, Statics, dynamics, orbital dynamics, thermodynamics, aerodynamics, fluid dynamics, chemistry and electrical. In short you will have more opportunities than most people realize. It will make you a better pilot because you'll understand more about the airplane and why it has what is has or does what it does.
I can tell you this since I'm almost finished with initial training. I had such an easy time getting through this training as compared to people of different walks because of my degree, things that make no sense to some, will make perfect sense to you.
As far as a concern for finding a job, I live comfortably in the fact that if for any reason I have to leave flying, (medical, family) I can get a very good paying salary it is a great peace of mind when I see some of the guys who have all their eggs into the flying basket, hopefully things will go well for everyone and will never have to worry about it.
Good luck with your future and don't let people ever discourage you from your goals.