How do the squirrels get into your run, Wendy? If they climb over the top of the mesh, maybe a mesh roof would exclude them ( and other more dangerous predators.) if they a getting in through the sides then so could rats, stoats or foxes. The only cure is to exclude them from the run so they cant get to rhe coop, and when you have worked out how to do this, your run will be safer and better proofed against other vermin.