While factually true, this does not defeat the point: those non-profit and public hospitals rely on paid procedures to pay their employees. And yes, those employees are furloughed and laid off en masse.
Government owned or Non-profit isn't as meaningful as you might think. End of the day, hospitals live on Medicare admissions for cash flow (ie. old people) and surgery for profit. Old people are staying home, people aren't hurting themselves on the couch, etc.
Government owned is only meaningful with the Federal government, which has unlimited resources.
Nitpick: Over half of US hospitals are nonprofits or government owned [1] though I believe hospitals are vastly outnumbered by private clinics and independent practices. Doesn't change your overall point - even the hospitals run by governments, the Catholic church, and other institutions have budgets and are watching their main incomes disappear without much hope for the near future.
[1] https://www.aha.org/statistics/fast-facts-us-hospitals