If you are going to be around all the time there is no real problem with using chicken wire for the run.
If you are not going to be around all the time you would be better off with something like 12 gauge 1/2" hardware cloth or even a dual layer to keep predators at bay. the problem with chicken wire for an "un-attended" run is that many critters can get through it quite easily. Raccoons, Opossums, Weasels, snakes, rats, mice, all can get through chicken wire only fencing with ease.
We have a run built for when we are at work during the week or on an errand run on the weekends. Our run has two layers currently and can have a third added if found to be necessary. The wire we have now is chicken wire with a 4' high wrap of 1/2" hardware cloth. The hardware cloth also has 2 feet under the ground, it goes about 6" down then out away from the fence. This is stapled down and covered with soil and grass so if an animal decided to dig under, it will hit the wire and stop digging. So far this has worked well for us. But if the coyote pack ever comes up, we can add an inner fence of 2" by 4" welded wire fence on the inside and we could even add hot wires on the outside. Our run is not completely covered but it does have crossing wires over the whole top for flying predator prevention, these are spaced to leave 1 sq.ft. open space between the wires.