Josh: I dont know how much experience you have with engineering as a college major, but there is more than enough overlap between Mech and Civil that either degree could be used for design. To be brutally honest, I believe coaster design involves about every major engineering discipline (Civil, Mech, Electrical, Aero, Materials, Envionmental, etc), so there is no one degree one MUST have to do the design, though I agree that ME would give the best skills set.
And Andrew, you cant get a degree in Civil or Mechanical (or Aero for that matter) without taking a class to demonstrate proficiency in drafting (by hand and computer)
jeremy