I live in northern Vermont. I'm currently building a coop and it will be insulated because our winters can get VERY cold. Our winters are probably very similar to yours. The best and cheapest way to insulate is with fiberglass insulation. It comes faced and unfaced -- faced is only slightly more expensive and is easier to work with. Where I live, a roll of faced R-21 insulation is about $40. One roll will do approximately 100 sq. ft.. R-21 is plenty good for a coop -- what's in the walls of your house in Nebraska is probably about the same (although what's in your attic is probably more than twice that).
If the walls of your shed are studded, and you buy faced insulation, you simply need to lay it between the studs and staple the facing to the studs to hold it in place. It's pretty easy.
Insulation board, such as Thermax, is very expensive. It comes in 4x8 sheets, equaling 32 sq. ft. For 1" Thermax, expect to pay about as much as you would for a roll of fiberglass. This means it's about three times as expensive, and won't insulate nearly as well. 1" Thermax will only give you an R-Value of 6.5, which is obviously much less than R-21.
Either way, you will need to finish off the interior walls to keep the chickens from pecking at the insulation -- either plywood or pressboard which is cheaper but kinda ugly.