If Mexico could get millions to self-deport, then we can too. Massive fines and imprisonment for those who hire illegals would go a long way toward turning off the employment magnet. If they can't get jobs here, they won't come here nor will they stay here. Not a complete solution, but a big first step in the right direction.
Actually, your argument is flawed seriously in two areas - one being the analogy between our situation and Mexico’s situation...the second is your idea to force deportation of those working ahead of those getting benefits. It’s much harder to do, penalizes our entire economy, and fails to recognize the abject pathetic nature of our own domestic unskilled and semi skilled labor force. Meanwhile, cutting off benefits immediately benefits all legal citizens financially, is much easier to do, and sends home the worst of the offenders.
Remember, any e verify system will be run by the same government that brings us Obama Care....so if you really want to trust in that kind of system, knock yourself out...