Actually, there might be two issues. One is the stain, but since most stains these days are now water based, that may not matter. It seems to me there are two versions of these stains, however. Some is mostly transparent and tints the wood certain colors, but you can still see the grain. The other is thicker and is intended to function like paint. I'd think the transparent would work better on the wet treated stuff.
Second issue is this wood may shrink, warp, twist and otherwise change shape some as it starts to dry. This may not matter on a chicken house. The stained stuff may change color some as the wood dries out.
Some are concerned about the wood being treated. As long as you are using the new stuff (tag on it probably says MCA somewhere), I would use it. MCA stands for Micronized Copper Azole. Copper dust is the preservative and it has been ground to microscopic sizes, put into suspension in a liquid and injected into the wood under pressure. The Azole is a fungicide.
These replace what was thought to be a more toxic treatment called CCA, with the A in this one standing for Arsenic, a poison. Not much, but some and enough the government decided it wasn't worth the health risk of people walking around in bare feet on decks made of the stuff. I would not worry about buying CCA. It is no longer available to mere mortals at the box stores. You have to find that stuff in special yards that sell to farmers for fence posts and pole barn builders and some commercial / industrial uses.
