I've heard that suggested before, and that particular idea doesn't sit well with me. It should not be the job of a private employer to investigate its employees to determine citizenship status. That's government's job, not the private sector's. And immigration laws should only inconvenience immigrants, not citizens.
A better approach, IMO, would be to keep the EEOC off the backs of employers, so that employers don't feel the need to beef up on their "diversity" ratings by hiring illegals.
I see your point. But still, if thirteen guys pour out of a 65 Volkswagon and apply for a job on your ranch in Southern CA, they can't speak English worth a chit, can't look you in the eye when talking to you and look like as if they've been sleeping in the desert for three weeks, you can probably assume they are illegals.