I'll speak from a Naval Aviation view point (which should be very similar to any services track).
You must have a degree (Army and maybe Navy WO flight programs exception).
It does NOT matter what your degree is in. There is constant debate as to who makes a better pilot:
the black and white everything has an exact reason and purpose technical kind of guy (engineers, mathmaticans, etc...) or the more free thinking, free floating, everything is the color gray type thinkers like music majors, theater majors or people who get a degree in Aviation
What track you are selected for depends on many factors including your performance (grades) in flight school, maybe even in what areas you excelled in during training to an extent, and certainly, and foremost, NEEDS OF THE SERVICE.

USMCFLYR