I don't think this is the problem but you may be using a mono cable. You need a stereo cable. The ends will have two plastic rings at each end.
It's probably just a faulty cable. Try another.
If your earphones play on both sides through the Zune, you'll know that the Zune input is not the problem.
If you know that you have a good cable, plug another device, like a portable cd player, into the car input. If the problem persists, the car input is faulty.
These things will help to point to where the problem is.