Of course they will. It will become just another entitlement program payroll tax. There will be no civil disobedient noncompliance because they'll simply confiscate your money before you ever even see it.
I'm convinced that the real beginning of the end for the USA happened whenever it was that the government gave itself the power to confiscate revenues via payroll withholding. This should never have been permitted. People should get their whole pay and receive monthly tax bills for which they would have to write checks to pay.
As long as the government has this power, nothing is ever going to change. We The People either need to start earnestly moving the economy under the table and off the books, or individual states need to stand up and defy the Federal Government and give their local employers cover to refuse to comply with withholdings.
Another possible situation is that you will see a functioning underground economy as you say.