Does taking Advanced Placement (AP) courses in mathematics and science provide students with more opportunities in the field of engineering?

absolutely!  taking ANY advanced courses shows your passion and your drive.  it will also make you stand out on applications.  go for it!

"More opportunities"? I would say no in general, though there are exceptions. AP courses give you a running start on college courses, and in principle they might allow you to graduate a quarter/semester or two earlier, but you'll be able to take all the same courses as part of a normal, four-year engineering curriculum, at which point the same opportunities are available. In practice I've never seen a company even consider high school work on a resume when hiring a new college grad ("NCG"); internships far outweigh even college coursework and grades. (Some schools like the University of Waterloo have "50/50" programs wherein students spend a quarter/semester taking classes and the next one doing an internship; by the time they graduate, students may have racked up four to six internships, at least in computer science.)

Of course, if you take AP courses in high school and do well enough in them to skip the corresponding college courses, you can instead take additional courses in college that you wouldn't necessarily have been able to in a regular four-year program. Or you could squeeze in additional internships if your college doesn't have explicit programs like Waterloo's. Or you could add a second (or even third) major; that's what I did. Any of those might indeed give you more opportunities.

You should also be aware that many companies will prefer students with a graduate degree (masters or even PhD, though usually masters suffices). For this path, your high-school work might be considered, but I've never been part of the graduate admissions process, so I don't know. They would definitely weight your college grades more heavily (and internships much less so) than companies do, and they almost certainly require standardized test results (GREs, at least when I applied) on which you might do better with your deeper and/or broader understanding of the subject matter.

But ultimately each stage (high school, college, grad school, job #1, job #2, etc.) is a kind of barrier, and companies may not look beyond the most recent one or two when hiring. So consider what your high school AP courses can do for your college options, and then consider what your various college options might mean for whatever comes next, but don't necessarily expect a direct benefit from your AP courses in your future job hunt. It's very likely to be indirect only.

As an aside, what you learn in mathematics and physics courses tends to be an excellent foundation for many fields. Again, the benefits are indirect, but analysis skills and problem-solving are fundamental to all engineering disciplines. Some of the more advanced physics labs (sophomore or higher) might actually have direct benefits, too. I took a year-long "electronics for physicists" course (Horowitz and Hill textbook), and it was one of the most useful courses I've ever taken, even though I'm a 100% software guy. Understanding how the entire "hardware stack" fits together, from semiconductor physics through discrete components to integrated circuits, actually can give you insights into CPU performance limits, error rates (yes, cosmic rays really are a thing!), I/O connectors (capacitance + resistance => RC signal decay; impedance mismatches => reflected signals, interference, etc.), SSDs, radio-frequency interference, and so on.

I agree with Zhoe, above, Martina. Successful completion of AP courses will help your college application stand out and will greatly improve your chances of being accepted to the college of your choice. I strongly recommend both AP Physics and Calculus for your senior year.

Good Luck, Pete Sturtevant, PE