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Same way a dog, raccoon, oppossum, etc. does - biting most likely. A hungry cat can be quite determined and quite innovative -might have to resort to squishing or kicking to open it but I have seen cats bite an eggshell and get it open.
If you don't mind having the cat around, start feeding it cat food (out of chicken reach!) - that might stop the egg eating. Sounds like the poor thing is quite hungry. If you don't want the cat around your place you can try to re-home it (there are some rescue groups that specialize in ferals...