I certainly agree that she is a game hen. And they are great momma's. The welsummer mistake seems to happen pretty often. And no not all green legged chickens are EE's.I've got a bunch of green legged games and none of them ever lay a colored egg.
As far as catching her goes your best bet is to be real patient and see if you can find where she is roosting at night. Just feed her a little late in the afternoon and watch her till it gets dark. Hopefully it's not too high off the ground. Wait till she's really settled in and its good and dark then sneek up on her from behind and grab her by the legs. She'll squawk like your killing her and try to fly away so hold on, then gently put her in a box and take her home. When I was a kid my Dad had lots of free range games and sometimes we would gather them up and thats how we would do it. But sometimes "wild" games will roost a lot higher than you might think.
Good luck!