They are incorrect, based on IRS Publication 926. When someone controls your hours, where you work, and what you are supposed to do, that makes them your employer. The IRS will not consider you a 1099 worker but rather an employee. The families that you provided services to are responsible for paying for ALL of your payroll taxes and their share of FICA, FUTA (social security, medicare, and any other payroll liabilities).
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