I personally wouldn't do it at all, but I think you could get away with it at slow speeds. The air resistance goes up with by the 4th power as speed increases, if my brain is working correctly at the moment... The air loads at 65 mph are way WAY higher than at 35. I doubt if the typical stapled splintery pine box spring would even stay together for long when subjected to those forces and strapped down tight. You would most likely end up with a busted box spring or possibly a bunch of kindling and springs in a bag.
Your insurance company "may" give you problems if it comes off and flies through someone's windshield, since carrying a box spring on the roof of a compact car is not generally considered safe or reasonable.
Think about it.
Don't you know someone with a pickup that you can borrow? Or just rent one from Avis for a couple of hours.
John E Davies
Spokane WA USA