MDF is probably the WORST material you could possibly use for something that is being designed as a knock-down construction. Unless you are already deep into the build, I'd say to just go with a good quality plywood. Then you can glue your batens and screw into those without worry.
Coarse drywall screws are great, btw. I've seen tests and the strength difference between the cheaper dry-wall screws and expensive wood screws is insignificant.