If I want to work at a tech company in the future in the area of speech processing / synthesis or something like that would it be better to major in linguistics and minor in computer science or major in computer science and minor in linguistics?
Take a look at (CS+X) Computer Science + Linguistics, UIUC.
As someone double majoring in these two majors, why not both? 
If you’re going to go with a major/minor combination, I would say to major in CS and minor in ling. It’s a lot easier to pick up on the additional linguistics stuff you may need than to play catch-up on the CS material you may have missed.
I’d look for a Computational Linguistics program first.
If you didn’t see any Computational Linguistics programs you like, then go for a CS degree and minor in Linguistics and/or look for speech or text recognition classes. You’ll certainly need to take a few statistics classes.