Are payments made to a health cost sharing network considered eligible to avoid the IRA early withdrawal penalty

Payments made to a health care sharing network are not payments for health insurance.  Since they are not payments for insurance, they are not deductible medical expenses.  Since they are not deductible medical expenses, they are not medical expenses that can potentially be used to avoid an early-distribution penalty on an IRA distribution.

Only the medical expenses not paid by your health care sharing network would be considered medical expenses.  

So, if you have a $5,000 "deductible" that you pay yourself, then that would qualify as a medical expense and would be applied as a deduction against your early withdrawal penalty.