Yes, you do claim the medical payments on your 1099-MISC for a medical expense tax deduction. Your deductions on a 1099-MISC would be reported on a Schedule C.
Any items on a 1099-MISC require the Self-Employed version (online) or Home & Business Version (desktop) of TurboTax to report. You will need to enter your health insurance premiums under Business
Income and Expenses in Wages and Income.