There are alternatives.
Samaritan Ministries is a non-Catholic health sharing organization. They distribute the costs of medical care among their members and negotiate fees.
Solidarity Health Share is a Catholic organization that will do the same when they get it going (so far, they have a signup form to request info when they become available). This organization was established specifically in response to the HHS mandate.
Strictly speaking, neither is an insurance company. However, the law allows participation in such programs
in lieu of purchasing insurance.