Unless your packed lunch is entirely nonsmelly and packaged so that it can be carried in surreptitiously, you will be unlikely to be allowed to carry it into the theater. A friend of mine had her dinner leftovers confiscated one evening when someone in the seat behind her complained. (This information is secondhand, but, I assure you, entirely reliable.)
So for those who are not privileged to use Pigmon's intermission delivery service....
The last time I did an extravaganza at Chicago Shakespeare ("Rose Rage"), they offered a box lunch option. I can't recall who catered it, but it was as good or better than anything else you can get in an hour on the Pier. (The fact that I was willing to eat it after all those abattoir scenes says a lot....) So you may want to see if they're offering that this time as well.
Otherwise, the best quick options on the Pier, in my opinion -- based on many years' experience although perhaps not in what Pigmon would consider the recent past -- are the Twisted Lizard in the food court (kind of a hike from the theater) or the Billy Goat. The Haagen Dazs place just by the theater elevators also does sandwiches, which are adequate -- just -- but handy.
If you have time to dine, Riva offers discounts to theater goers, which bring their food down out of the hideously overpriced range. It can be very pleasant to sit in their dining room overlooking the lake (and you'll do it awhile, because service is sl-o-w). Avoid anything complicated and stick to the simply prepared seafood options, like oysters or grilled fish, and the food can be quite good (based on my extensive, almost-recent experience and conversations with many theater-going friends and aquaintances).