I think it is a completely different subject. One is the problem of illegal immigrants be they lettuce pickers or Richard Branson. The other is the problem of the working poor legal or illegal. You solve the two independently. For even if you cut back on government benefits to the working poor to encourage greater self-reliance, you would still have a large number of illegals crossing the border to get jobs here that they can't get in Chiapas. A $6.00 an hour job in San Diego is better than eating prickly pear pads in Tapachula.
They're not going to cross the border for a $6 job in San Diego if you crack down on employers so none will offer an illegal a job.
How do you do that? Simple. Set up an instantcheck system for legal employment status verification at the DHS. Any employer who fails to utilize the system and hires an illegal gets severly penalized: $100,000 fine plus jail time. Possible seizure of assets under RICO if it's not a first offense.