"Right" is perfectly fine, or we could say "the right way", or "correctly", etc.

I assume you mean your son put the right shoe on the left foot and the left shoe on the right foot. You would say, "Your shoes are on the wrong feet." It would be impossible to put them on backwards because the heel of the foot would be in the toe of the shoe. You would tell your son to put his shoes on the correct feet, to make it clear. The word right has several meanings. In England, one drives on the left side of the road. That is the right side of the road in England. In America, one drives on the right side of the road but that would be the wrong side of the road in England.

Put his shoes on the wrong feet. Put them on the right feet. Put them on right. Put them on correctly.

Yes, and it confuses us occasionally. "Could you put them on right?" is absolutely correct and it is what we would normally say. However, there are situations where you do have to be careful. For example, if you are in a car and the driver says "I think we should make a left turn here", don't reply "right." Instead, reply "correct".