I have to dis-agree---if that's OK. The quickest way is to snap their neck or as I do---hold them by the legs and quickly sling them downwards hitting the back of their head on a solid table(or similar). Lights out!The quickest way is Drowning in warm water..Sounds brutal but its quick....Only use that method on just hatched Chicks though....
What a useful post! Even tho late for the Original Poster your reply will inform so many others.I actually haven't seen any other posts about guts hanging out, but I experienced the same thing with the first chicken I ever hatched!!! I did not stitch her up though. Instead I gently pushed the viscera back in, and used Steri-Strips to form an 'x' over the belly button. Then I wrapped her in a warm wet paper towel and put her in a cup (on a heating pad, under a light), and periodically applied neosporin to the skin around to keep it lubricated. I fed her probiotic water and eventually some water/food mash from a plastic syringe for awhile also. I kept checking on the strips and would re-apply as the hole began to close.
However, I did recently come across this article recently and it seems to be a good one to have around just in case:
http://www.alysion.org/euthanasia
Helium is also a good option, because there is apparently less distress and is advocated by groups that believe in the freedom to choose a peaceful dignified death, like 'Last Exit'. I imagine a couple of helium balloons would work for this on a small animal.
I know just how traumatic the hole in the stomach issue is!
Great job being brave and doing your best to help them out.
(I realized 1/2 way through that this was from March of last year. I wish I was there to assist when it mattered. I am so sorry)
Thanks for taking the time Hatchichickens 
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