Seems to me that all you have to do is establish an employee data base the employers have to verify status with.
Make it Federal, consisting of a picture, SS or other number & DNA in case of criminality.
Make the employers verify the person and if they don't and pay under the table, fine and eventually award the employer with a felony.
Do that and millions will have to go back on their own first to gain legal entry or they will have little or no work,
The illegals will have to remove themselves or probably starve and we won't have to use resources to remove them.