Do I need an undergraduate degree in engineering to apply for?
You should expect your coursework to be quantitatively oriented in MS&E, it is in the Engineering School. A full mathematics and computer science double major or major/minor would still find the material challenging depending on the undergraduate institution. You want as much engineering background in math and CS as possible to make your transition to grad studies easier. I'm pretty sure you need at least math up to all of calculus and linear algebra, and cs with intro programming and data structures for basic preparation. Good luck.