I store mine in boxes that I got from IKEA years ago. They are sectioned, like boxes you would store Christmas ornaments in. They no longer sell the exact boxes I have, but they have lots of storage boxes that would be suitable.
Another box I like is the clear plastic box made for holding scrapbooking paper. These are available at JoAnn's, Michael's and Hobby Lobby or most any craft store. You could have one for each color or type of squares!