The stock bumper bolts to the end of the frame in front through two holes slotted through the body. There are two bolts per side, accessible through the fenderwells. There are two holes per side you can see the bolt heads through.
1/8" should be good for the skin, yeah. I'd think that 1/4" plate would be good to use for the attachment points and internal bracing. With a 1/8" skin all welded together, it'll be fairly rigid all by itself. Seems like perhaps the square stock may take up a bit of interior room in there and you'd still need some plate to bolt it on with anyway. For bracing though, the square would work fine.
I've been thinking about doing the same thing, I just need more plate and the time and money to do it. I'd like to actually weld in some better frame horns or mounts for the upper mounts, that way you could bolt the new bumper to the frame horns or mounts with them sticking out of the body already and not have to deal with positioning a heavy-ass bumper by hand to fit the teeny bolts/mounts that were holding the stock bumper on.
I'd definitely like to see what you come up with. Remember to add recovery points and some bracing or plate reinforcement for them so you can use the winch and a pulley on yourself, or just for towing Jeeps out.