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What critters do you contend with? Some folks here have to worry about large dogs roaming their property, bear, bobcat, etc. The other factor is your security strategy. I used 19 gauge 1/2 by 1/2 inch wire for the run, but 16 gauge 1/2 by 1/2 inch wire to cover the coop's windows and vents. Our yard is fenced with a solid board fence, so neighborhood dogs won't likely get in, and we don't have any larger predators to worry about where we live (thankfully). At night, when predator risk is higher (raccoons, for example), our chickens are locked inside the coop.
19 gauge wire is thin enough for me to bend with my hands and to cut easily with tin snips. I think a very determined critter with time and teeth could probably gnaw through it if it could get its grinding teeth into it, but I think the small openings like 1/2" by 1/2" would make that difficult to do (that's my theory, anyway). I wouldn't use anything thinner than the 19 gauge, personally.
The 16 gauge wire is very tough stuff. I can't bend it, and it's an awful job to cut it with tin snips. I would be amazed if any critter could chew or tear through this wire.