I've been trying to find the time to thoroughly research these alternatives to health insurance but the end of the year is nearly here so posting some references and a brief introduction to these will have to suffice for now.
While we don't require patients to have any insurance to join our practice, we recommend a high deductible policy or some other means to protect against large medical expenses as well as the ACA (Obamacare) tax penalty. However, a large minority of our patients still find their insurance options too expensive and don't even have catastrophic coverage. The Health Care Sharing ministries are a less expensive alternative to insurance that provides some cost protection and four of them are reported to meet the requirements for avoiding the ACA tax penalty.
So far, we have a handful of our patients who use these these ministries but our experience with them at this point is limited. They are not insurance policies and differ from insurance in how they are regulated and how (and which) medical expenses are reimbursed. They don't restrict patients to narrow provider networks like HMO insurance plans. Most of these seem to require members be Christian but one, Liberty Health Share, has liberalized its membership criteria to a broader set of beliefs and is not restricted to Christians.
US News and World Report 2014 article
A 2013 NPR story about Medi-Share
A Feb. 2015 progressive opinion from the NY Times
An article about CMF Curo, a Catholic version of these
A critical 2011 Consumer Reports article quoting former Maine Supt. of Insurance Mila Kofman