The article proposes that everyone (healthy or sick, young and old) pay $350/month out of their BI for health insurance. Would that be enough to replace Medicare?
Sure, but all you've done is keep medicare in place and reduce BI to $7800/year. Note that medicare is only available to people over 65 in most cases - it isn't generalized health insurance, so now anyone under 65 needs to pay for health insurance on top of the Medicare tax. Which is exactly how it is now anyway.