Good afternoon, It is illegal to keep wildlife (squirrels included) without a permit. Wildlife rehabilitators are issued a special permit from the Illinois Department of Natural Resources that allows them to provide temporary care for wildlife. Their goal is to rehabilitate wildlife to the point where they can be successfully released back into the wild where they belong.
An animal in a cage, no matter how well it is care for, will never experience the freedom and quality of life that a wild animal enjoys. It is great that you've been feeding the squirel walnuts, one of its natural food sources. But imagine how much more fun the squirrel will have using its brain to find its own food and finding clever places, like flower boxes, to hide its winter stash of walnuts.
That being said, it is in the squirrel's best interest for you to release it where you found it. Cold weather will be coming soon, and the squirrel needs time to build a nest and store food supplies for the winter. The longer it remains captive, the less chance it has of adequately preparing to survive the winter. Despite the fact that it seems tame, it knows how to take care of itself. (Though it sounds like it probably appreciated some help with the bluejay...they can be bullies).
Another concern with a wild animal that appears tame is that it might be sick. This is another reason that people should not handle wildlife.
I would recommend releasing the squirrel early in the morning. Do not approach the squirrel once its been released. It will soon get back to its business. If you have more questions, please let me know. Thanks for writing.