No, you cannot store a box at an airport anymore. You might be able to find a storage company in your destination town that will take them, or perhaps relatives or friends.
However, if none of that works for you, my suggestion would be to use a cardboard box, which you can get for free
from your local LBS. When you get there, you can either toss the cardboard box, and ride out of the airport, or simply fold the cardboard box up and take it with you. I've done both.
When you are ready to leave, you can either hope for an available cardboard box from an LBS there (that worked for me in Australia), or that the airline sells or gives away cardboard boxes. Call the airline, don't just assume. From my experience, Air Canada does not have boxes, but United does for $15, and Virgin Blue (within Australia, which doesn't help you right now) does too, and gives them away for free. So when they are available, they aren't overpriced at all.