Chicago public restrooms
The malls (Macy’s, Bloomingdale, Nordstrom, etc.) all do have restrooms, but they tend to be hidden on upper floors or in basements. Ask a security guard or sales person if you have the need.
There are very nice restrooms in the basement of Macy's (the former Marshall Field's) on State Street (one block east of the Daley Center) , and there are also lovely restrooms in the Chicago Cultural Center (two to three blocks east of the Daley Center) at Randolph and Michigan.
Bookstores always have pleasant public restrooms, usually on their upper floors.
Walgreen’s on almost every corner of Michigan Ave have restrooms, just ask.
Every fast food place or coffee shop has a restroom (again usually upstairs). The fast food places often lock the restroom doors, so you have to ask for a key at the counter. They may prefer that you're a customer (and buying a cup of coffee never hurts), but legally they have to let anyone use them.
All the downtown train stations (as in Metra, not the 'L') have public restrooms.
The same place that you would find them in any big city; adjacent to the lobbies of the big hotels.