We did ours with 4x4 posts (we used round ones on the run) and 2x4 cross pieces. A 2x2 would probably work ok, but the 2x4 makes it very sturdy - no worries at all about this thing moving or falling apart.
The 2x4 was easier to staple the wire to, also. You have more surface for the staples. Also - being as how it is outside, the 2x2 might warp more easily than a heavier piece of wood. Warpage is bad
Treated wood will help - but it isn't a guarantee of non-warpage. I have 2x4 treated in a fence out by the barn that warped - all treated wood is not created equally. If you paint, you will have to paint more than just that first time. Possibly once a year, but if you're lucky maybe only every two years. I try to never paint anything like that. I doubt the chickens would peck the paint off (they might, I suppose) it is more a maintenance factor in not painting.
You can see my run if ya click on the webpage thingy below my name and pic (to the left). The run is on the main coop, (and the main page).
Oh, one more thing - just a personal observation, take it how you like .....
I did the wire top thing. It was very hard to tack that wire (and get it tight and straight) and it was not fun. I ended up covering half (after I had already wire tacked it) with a roof - so they could go out in the rain and have a dry place to be.
This summer I ended up finishing covering it in roof. I could have saved myself putting that wire on the top in the first place if I had thought about the mess it would make when it does rain (we don't get much snow here in winter, just a lot of rain - but snow would eventually melt and cause the same problem).
Now, I also put in sand on the floor of the run, and that helped - I cannot recommend highly enough putting sand on your run floor. It still gets wet in there a little (sideways rain) but the roof keeps a lot of it from pelting the poor little buggars when they want to be outside and the sand helps soak up a lot of it, so saves on the muddy poopy slip and slide out there.
They get plenty enough sunshine coming into it with a run that long and tall, and they prefer having some shade when it is too hot and sunny, anyway.
We just used some old corrugated metal sheets that we bought for a buck a piece for the roof. We caulked the old nail holes, and it works like a charm.
I would absolutely not use anything smaller than 4x4 for run posts. Not just because of strength but because of longevity -- thinner wood, e.g. 2x4s, will much more quickly rot thru to the point of being too weak.
Pressure treated wood (or cedar) is an awfully good idea for posts being set into the ground.
Pressure treated is actually IME *more* apt to warp than regular SPF, however it will last vastly longer if it's in contact with the soil.
For crossbracing I would use 2x4s again for durability reasons. If you are on a strict, strict budget and it doesn't have to last forever, and have the tools to rip 2x4s down into 2x2s yourself, you *could* use those 2x2s for your bracing... but for heaven's sake don't BUY 2x2s for bracing. Because they're so ubiquitous as studs, 2x4s will be cheaper than any usable quality 2x2s you can buy at the lumberyard. I'd really recommend 2x4s though.
just my 2 cents, Im saving up for a 12x7x6 chain link dog kennel from Lowes, $200. Im not that handy with framing and I think the lumber would cost just as much. I wish I could have one of those nice wood framed runs, mine looks a little redneck right now.
Hmm would need 4 4x4 8 foot posts, got them under the deck from the deck redo last year, will need to frame up some 2x4's and staple chicken wire to them i guess??? and then a roof over the first 6 feet or so and leave the other 6 feet open with wire top?? or roof the whole bloody mess lol...