Chain linked isn't the appropriate fencing for Button quail. They do best indoors since they are so tiny. They are native to subtropical forests of S.E. China...they do best in an average of 70 degrees F. They are also monogamous (one male per female) and should be housed in pairs.
I don't have experience, so maybe someone can correct me if I am wrong. A friend who used to raise quail told me that newly hatched button quail could escape his inside pen, made of aviary cloth (1/2 inch chicken wire). The pen could contain adult button quail, but the babies are that small.
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i don't have Buttons but i understand that when they're babies they are only the size of Bumble Bees so figure that if a Bumble Bee can get through it so can a Button baby...
They can fit through chicken wire, you best to use hardware cloth for their cage. They do not do well in drafts or outside in the winter. They do best inside.
Now my coturnix can not fit through chickenwire and they do well outside if kept draft free
They also are less flightly and let me pick them up.