Boxer primers have a single central "flash hole" under the primer like virtually all U.S. commercial ammo. This means your reloading setup can punch the old primer out with a single centeral pin. Berdan primers have (I think) two non-centered holes. The brass case blocks the center of the primer. I think it makes sort of an "anvil" for the primer to hit against. The problem is, your reloading setup won't be able to knock the old primer out of the case (and may be damaged if you try too hard). I think you can get some sort of a "claw" device that grabs the primer externally and lets you pull it out, but it probably wouldn't be worth the trouble unless you were reloading some rare old cartridge that was only available with Berdan primers.
As far as inexpensive .308 ammo, I use the British Radway Green ammo that I bought at a gunshow for (I think) $60 per 400 rounds in a GI ammo can. Its Berdan. No good for reloading, but has the benefit that I don't feel guilty for leaving/losing brass.
I almost went with Hirtinberger .308 ammo. It has a good reputation, costs slightly more, might be a little older (at least what I looked at was), but is Boxer primed and reloadable.
I'm no expert on .308 ammo and can't comment on the others available. Good luck.
Doug