Chicken Wire's sole value is keeping chickens in.
ALL of your predator deterrent will come from your electric fence strung around the outside. ALL. If that's adequate for your local predators, and you have no concerns with snakes (or the birds you are putting in the run are too big for your local snakes), you are good to go.
If you have digging predators of small stature, or big snakes who can fit thru chicken wire (and a snake can swallow something many times its empty diameter), then at minimum you need to hardware cloth above the height of your first hot wire, and extend a skirt around the cage. Or accept losses (like the snake that got in, but can't get back out).
And if he's custom making them? Offer to provide him the hardware cloth as alternative to is use of chicken wire, have it "custom" built with a materials substitution.