Apparently over regular trade, yes. Wonder Trade doesn't allow it. The Original Trainer will be the person who has them when they hatch, so that means it'll be like he's the one who caught them.

He says if it's not too much trouble if you could give him a few of the eggs if that's possible so he has a better chance of getting a decent nature. All he has to trade are a few of the basic route 1 things.