If you can make a support frame to hold up chain link on top it's probably your best bet to protect from climbers and birds of prey. We have birds of prey here on and off throughout the year, the ones that stay the longest are Coopers, which I wasn't that worried about until a neighbor 5 miles...
10x10 = 100 sf, which is > 10 sf per bird, so I think it's enough run space. Chicken wire will not be effective as a deterrent to anything wanting to get in. We have our birds in chain link runs with chain link across the tops, used top poles and assorted other hardware to make supports across...