Your hutch looks pretty secure, what type of hardware cloth are you using? Do you know that it is made of? If it's stapled in you can totally put TONS of staples on the sides where it meets the wood and see how that goes. Are you keeping your hutch right where it is now? I'm not an expert on the raising of quails, but if you want you could move it closer to one of the buildings on your property so it's not so wide open and less of a target.
EDIT: oops you said what kind it was in the first post lol you can totally do 19 ga, and you can totally go bigger or smaller, i knew a person who built their coop with 24 ga. it was hard to work with, but effective. i'd recommend 14 if you want to be really safe, but because it's off the ground you've got much better chances. also try to get smaller holes in your wire just to be safe