Yes a bobcat will easily go through a chicken wire fence, chicken wire is not strong at all, it is made to keep chickens in not keep out animals trying to eat them.
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Just a thought on the eating, have you tried giving them something squirmy like a worm? Chickens love stuff that moves. The dog can be helpful but there is no such thing as a flock friendly dog, that is achieved by many hours and days and months of training and some dogs will never be completely trustworthy.