The following is purely anecdotal evidence, so feel free to take it with a grain of salt. It only covers one year's class of CSE students.
While there are a number of some science degrees that are acceptable, I highly recommend getting some kind of engineering degree if you wish to become a CSEO. During our apps course (The 8 month long ashore training period where you learn the theory behind everything) we had a number of students who had some of those "other" degrees, alongside the rest of the class with engineering degrees. And I can definitely say that the students who had a non-engineering degree struggled much more with the material. Especially those with Comp Sci degrees. As far as I can tell, the main reason behind this was the lack of math skills (Or at least what I perceived to be a relative gap in math skills), Calculus in particular.
I'm not saying that everyone who wants to be a CSEO needs an Electrical Engineering degree. I've got a Mechanical Engineering degree myself, and pretty nothing I learned in University past 2nd year is even tangentially applicable to my job. But, like all engineers, I also got Calculus up the whazoo. So this allowed me, when taking a crash course in Electrical Engineering (which is pretty much what 90% of the apps course is), to not also require myself to be taking a crash course in Mathematics as well.
So if you are going to be doing some non-engineering degree, and are looking to get in as a CSEO, I highly recommend you take some higher level Calculus courses as options. My :2c: