Co-op is the word you’re looking for. These co-ops went far beyond just healthcare after all. They owned bakeries, pharmacies, entertainment facilities for the workers, …
There isn't really a good translation, the concept doesn't really exist in other countries. They have aspects of unions, co-ops, governemental and political organisations. In the past, they had their own legal status. Today, the EU forced them to become real corporations, and e.g. had them split of the travel aspects from the health insurance aspects.