The one acquaintance I had who was employed by such an off the books care provider found that upon being fired the employer was not part of the system and initially her UC was denied--but upon pressing the point she was entitled to collect and the state proceeded against the employer. I say this in context of do NOT quit if you expect to qualify for UC--hang in there long enough with no whistleblowing to work long enough to qualify.
If wife wants to make noises and blow whistles --then do so after she has been there long enough to qualify for UC.
Whether or not your employer gives you a W-2 you need to pay the state and Feds the proper income taxes due and that means any estimated taxes as well unless you otherwise fall into a safe harbor. If you are trying to stretch out the date of noise making consider filing for a extension of filing your tax return.
I do not know the mechanics of what happens if employer messes it up but I suspect you still owe all taxes that employee would pay on income but that employer bears problem to make up the gaps as to stuff employer must pay. EG you pay your share of SS as an employee--not the higher rate as an IC.
In theory I think there is a small reward to turn in a federal tax cheat and provide usefull information to support same---but I'd not assume its a big reward........or that it would result in major pain to violator.
DO NOT engage in threats of pay me or I will turn in your illegal conduct to law enforcement--such a threat can become a form of extortion--and backfire!