True..only one or two would put the scare into the employers and dry up the jobs but what about that slight complication of them not having any rights as a legal citizen to sue in a US court?
That idea would never dry up the desire for illegals to cross the border. What you'd do is penalize the US worker through loss of employers via lawsuits, the demand must be stopped by a target effort of removing incentives for illegals, not by adding them.
Remove all wage protections that aren't adjusted to the illegal's home country's exchange rate, force employers to offer jobs to legal US workers likewise adjusted (an $8hr job for an illegal Mexican ought be $86.24 USD/hr for a legal US worker) and quickly the incentive between employers/illegals ceases.