People work without documents for a variety of reasons, mainly they are trying to avoid scrutiny, pay withholding, or they are in the aftermath of a lifewreck of some sort. Some are criminals, some are tax scammers, some are just at the bottom of life's barrel.
The economics of a minimum wage can be argued both ways, the whole freedom of contract argument is not exactly bulletproof because the labor market for day laborers or other lowest-level, low skill workers is not a true free market situation since all the bargaining power is in the hands of the employers, it's more of a 'take-it-or-leave-it, I can probably hire an undocumented guy even cheaper' deal.
There is a minimum wage law and an FLSA and that isn't going to be changing in the foreseeable future, so what anyone thinks the rules ought to be theoretically isn't really germane to the SecLabor's statement or the current real world employment/illegal-immigrant dynamic.