The short answer is no ... you don't need a 4 yr degee to work at SWA, but it helps put you in the majority. Some airlines have the 'catch all phrase' (4 year degree preferred from an accredited college/university).
My new hire class at SWA had the following demographics:
24 in class
Ages : 25 to 51
TT: 2025 - 14,000
Civil : 50% / Military: 30% / Civil/Military: 20%
5 had little (less than 100 hours) jet time
Civil : 3 FAR 91 (DA-50, LR-JET, GIV,V) / 3 FAR 135 Scheduled / 2 Unsched FAR 135 / Rest were FAR 121 Jet / Turboprop
Military : Almost perfectly split between single seat (and RIO capable A/C), transport (C-5,C-130,C-141) and heavy Bomber (B52), 1 Astronaut, 1 U2/SR-71 driver, 2 from the ATC (Air Training Command).
100% of the military guys had at least a Bachelor degree, 75 % of them had a masters (all over the map ... Mathematics, Physics, Electrical / Mechanical Engineering, Economics).
95% of the civilians had a Bachelors degree, 25 % had a masters degree, 1 had a Ph.D. (again ... degrees were all over the map ... Mathematics, Engineering, Computer Science, Information Systems Management, Economics, Finance, Chemistry)
The military guys made up 80% of the first half of class (older group) ... most were retired 05s and 04s still serving in the reserves.
From my records we did not have an 'Aviation Science' major in our group. I don't know exactly what that degree entails but from my perspective I agree with a prior poster ... get a degree in something you can use God forbid something happens to you to disqualify you from flying.
So in a word or two ... pretty varied backgrounds. Our class meshed very well. We had little or no clique action going on. We studied, drank and had dinner together.
Sorry for the run on ... I was amazed I could find those records from class actually.
Best of luck ...