No. You cannot classify wages paid by medicaid/medicare as schedule C (self employed) income.
In fact, most such wages are excludable from income (but are still eligible for the Earned Income Credit).
References and details on how to report:
https://www.irs.gov/Individuals/Certain-Medicaid-Waiver-Payments-May-Be-Excludable-From-Income
https://ttlc.intuit.com/turbotax-support/en-us/help-article/form-w-2/qualified-medicaid-waiver-payme...
https://www.publicpartnerships.com/media/bylb5fvj/difficulty-of-care.pdf
https://www.irs.gov/businesses/small-businesses-self-employed/family-caregivers-and-self-employment-...
