You can't assign them an armour-only uniform, because the pick / axe / hunting gear is a uniform and overwrites any you would assign yourself. At least they wear something now instead of running around with only their pick to protect them, which makes a skilled miner a little less of a glass cannon than before.
I was afraid of that. In the past, before I knew it was a problem, I had marksdwarves who were hunters and I didn't experience much trouble with the uniforms, and for some reason I seem to remember them wearing their armor while hunting. Might have just been a case of the dwarf having only partially switched uniforms. Oh well, making leather clothing should help at least.
Noob question here: by "equip civilians" you mean creating peaceful squad(s) (soldiers you never send to fight, nor train) and set up armor as for regular squads?
Yeah, that's the idea. You put civilians in a squad, assign them a uniform with as much armor as you want, and maybe a weapon, but never set them to train. That way they'll be wearing some armor if an ambush or thief pops up beside them. It won't make a world of difference, but should help. Unfortunately, as was just pointed out, hunters, miners and woodcutters don't work with this because the pick, crossbow and axe are parts of a hidden uniform that the dwarf will change into when they go off duty (i.e. all the time in this case), so they won't wear the armor or pick up another weapon correctly.