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I think it depends a lot on your situation. If you expect to have packs of hungry dogs or professional raccoons rippin' at your windows, I'd go with the heaviest gauge available and attach it real well. In a normal environment I just wouldn't use anything super flimsy.
I am not sure how worthwhile it is to use hardware cloth to bury or use as an apron around the run, btw. All that will help against, that 1x1 or 2x4 mesh *wouldn't*, is weasels and rats -- and quite honestly, if you have a weasel or rat problem, they're likely to find a way in no matter WHAT you do and the chickens just need to be locked up at dusk. (edited to clarify: I absolutely would use buried wire or an apron [I prefer the latter], I'd just use 2x4 wire or 1x1 if I felt like splurging)
Good luck, have fun,
Pat