I like this idea. The second part, well, it's a bit odd, and I don't know that MC/Visa/Amex would actually do it, but, I get his point on the whole thing. However it might just be a thought experiment intended to do something about the problem.
If you look at the Gingrich/Cain debate on YouTube he explains how FedEx can track billions of packages in real time and you can log in and see where your package is in seconds. He wants to use the technology that businesses have been using for years to streamline and solve government processes. This could be used, for example, to see if people are here legally when they apply for government assistance or private sector jobs.
Of course they would. They'll get millions to perform something they already do.
It's not surprising Newt wants to privatize that part when he's in favor of privatizing the guest worker process. A monumental mistake IMHO.