**Note--I looked back through this, and I'm way off topic for the purpose of this forum, which is breaking into the industry. If you moderator folks feel I'm too far off, I can edit it all out if you'd like.**
Well, I may as well throw a couple of cents in here, hopefully they will also make sense as well.
I have an engineering degree, and I'm currently back in school in a post-baccalaureate program in computer science that Oregon State offers. The reason is that I've found that I don't particularly enjoy the engineering work that I'm doing. I do seem to really like programming, so I want to make that switch.
Don't get me wrong, the engineering work isn't terrible or anything, and it does pay pretty well. The thing is, like with a lot of jobs I suppose, I spend a lot more time in meetings and shuffling paperwork about than I do actually designing test fixtures and setting up production lines and fun stuff like that. Now, your mileage may vary, of course. You could end up in a job in which you don't have as much paperwork overhead and you get to spend more time doing the fun part.
From what I can see, it looks like computer science offers a lot more flexibility in your day to day life than engineering does. This may just be because of where I live (San Francisco bay area), but there is an absolutely huge demand for computer science. The jobs pay very well, and they exist pretty much all over the place. If there's a particular place you want to live, there's probably work there in computer science.
As far as engineering goes, mechanical would be a good choice because there is a lot of work available. Electrical also offers a lot of work, and it has overlap with computer science. When I look at job postings, companies actively attempt to recruit programmers from other areas, and are willing to pay them to relocate in some cases. That doesn't tend to happen in engineering. Also, the engineering jobs don't pay more than computer science jobs. I can't speak to pay rates in game dev computer science jobs, since I'm not in that field, but if you're looking to get a high salary to eventually save enough to start your own business like you say, engineering is not more lucrative than programming.
In any case, I'm not sure from your last post whether you're still considering a CS major with a ME minor, or switching to ME entirely. However, since you already program on the side, it seems pretty apparent that you enjoy it. If you're looking to switch to ME entirely for the money, I'd reconsider it since it doesn't pay more than computer science. If you're looking to do a minor, because then you get to learn more physics and engineering and such because it's stuff you really want to know, that sounds entirely reasonable. However, as others here have pointed out, there's no need to decide that immediately. You can get into school and experience the workflow there a bit before you really need to make that call.
One final note, if you do want to open a studio, there's actually a lot of other knowledge that would be very useful to you. This is stuff that would happen in the future a bit, but getting an MBA would probably be a wise choice. There's quite a bit involved in the business side of things that can be easy to overlook but can make your studio fail.