Possums LOVE eggs....get ya some either fresh or grocery store or even rancid out of the incubator didnt hatch eggs and put them in a live trap with some stinky canned cat food. Last time I was in
TSC they had 2 traps together on sale for about 34 bucks...not a bad price at all. Put them close to your house or coop or anywhere that is dark during the day....possums LOVE to live under houses, sheds etc when its light out and they come out at night!
As for your pen....with that kinda issues (coytoes-we have them here in FL too and they WILL chew threw a large wire pen to get at something or dig under) I suggest small wire cage/pen...harder for them to rip apart with their teeth. Chicken wire is nothing for a coyote and coons reach right thru it. You can also sink your wire DEEP into the ground at least a foot...and if the ground is frozen...you just make a wire floor and/or do a generous apron around the pen till you can dig your ground.
Always make sure (besides a coop) that part of your chicken run has 2 solid corners....that way if a coon reaches through and gets one...the rest have a protected corner to go to.
A fence around your property is a GOOD THING as is a chicken loving noisy dog.
Also other deterants....can get cheap battery operated bike reflector blinkers and turn them on at night. Any type of light in the coop/pen will deter predators. Also you can string some CDs on fishing line...they blow in the wind and throw off reflections that deter predators. Hot wire is GREAT...you can get a solar operated hot box for about $60 new and then the cost of hot wire. That will deter most anything. Ihave a friend in Chicago area-outskirts who ties small treble fishing hooks about 3 inches apart and strings them around his peremator....one end tied to a can with bb;s in it...that way if it snags something and it runs he knows which way to shoot....but dont to that if your chickens are let out as it will snag them as well.
Fire also deters the critters of the night if you can have a small burn pit and burn dead branches etc a few times a week after dark and let it smolder.
Now I am in FL...not Illinois...but many predators are common all over the US, esp coons, possums, foxes and coyotes. Given my choice I hate the racoons the most as they come in packs and are the most clever. So make sure you have a good roof/covering as they and possums will climb t the roof to find a way in. Hope this helps!