Ok. So back on topic then…

Are non-Catholics not allowed to touch a Catholic after they have taken the Holy Eucharist during “Our Father” because the Catholic would be made unclean if they made human contact with a non-Catholic?
No, not at all. We Catholics aren’t in the business of judging what’s clean and unclean. Like what Jesus said, things of that sort just don’t matter – what happens to your body doesn’t matter; the only thing that matters is what your body does. The Old Testament talks about clean and unclean objects and gentiles, but we understand that it was just a part of the Covenant until the Messiah comes. The Messiah came; the Apostles weren’t sure if those cleanliness laws still applied. Our first Pope, Saint Peter, decided that we no longer need to worry about the clean and the unclean. This is why people like Mother Teresa, John Paul II and Father Damien of Molokai got to work with lepers. They, if anyone, would be considered “unclean,” but Jesus sent us to be with everyone who needs us. As Christians, we understand now that no human being is unclean. “What God has made clean, no man has the right to call unclean.”
With the Our Father, there isn’t supposed to be holding hands at all, Catholic or not. The rules just specify that the hands are supposed to be raised – no holding is supposed to be involved at all.
Like I said earlier, it really is only in America and America-influenced countries where the custom was picked up. It’s actually improper.
Take a look at, for example, Masses with Cardinals. There are videos of them on the Vatican website. If any living people are “clean” these days, it’s those men. But they don’t hold hands at the Our Father either.
And *after *the Our Father, we are invited to offer signs of peace to the people around us. The sign of peace can involve a kiss, a hug, a handshake, whatever; we’re called to share the peace with those around us, Catholic or not. The person could be a satanist for all it matters, but we still are invited to share peace.
After Communion, there isn’t any rule about touching anything.
Why ever that boy didn’t hold hands with you and your daughter at the Our Father is his own thing and it doesn’t really involve any Church customs. Perhaps it was because he was shy. I’m a 23 year old guy and I can tell you, it can awkward to have to be with your date’s parents. Maybe he was so nervous that his hands were cold and sweaty.
Or like I said previously, it may just have been that he knew the rules that you really aren’t supposed to be holding hands at the Our Father at all; he might have just been following what he knows were the rules.
Whatever the case, if I can give some advice, maybe it isn’t a good idea to be so alarmed by his not holding hands with you. It doesn’t seem like it was a sign of rejection – if he were rejecting you and your daughter, he wouldn’t have had brought you to Mass with him. If you really wanna see bad, you should see what I do when I want people to go away.