I am harvesting all that are mostly orange. The ones that are fully green or more than 50% green I am leaving. Kind of depends on the vine. If the vine is healthy and the pumpkin is green then I would leave it as long as you can. I have SVB, squash bugs and cucumber beetles AND some other small critter chewing on the fruit. So I am getting in what I can to save it, but the mostly green ones will not ripen off the vine.
I have put up an electric fence to keep the deer out after they started eating everything last year.

Your mention of chicken wire reminds me that last year before I put up the electric fence I used chicken wire cages that we built and used to use to protect out new trees at our previous house. Worked great but just had too many to cover, thus the electric fence. Doesn't keep the bunnys, chipmunks or mice from getting to them.....I live in the woods.