I’m seeing this for the first time and most of the above answers are wrong.
There are two separate issues. Contributing to an HSA and spending money from an HSA.
Spending is simple. Once you have money in an HSA, you may spend it on any qualifying medical expense for yourself, your spouse, or your dependents. You can spend HSA money even if you are no longer enrolled in a type of insurance that makes you eligible to contribute. Qualifying medical expenses are described in IRS publication 502.
In order to contribute tax-deductible funds to an HSA, you must be enrolled in a qualifying HDHP—high deductible health plan—and have no other medical coverage that would disqualify you. For example, if you enroll in an HDHP at work, and your spouse is enrolled in a different insurance plan that provides back up coverage to you, then you are not eligible to contribute to an HSA. Or, if you are enrolled in a single HDHP at your job, and your spouse has single insurance at their job but also has an FSA, then the FSA disqualifies you because even though it is not in your name, your spouse can use it to pay for expenses for their spouse (which means you) and so you have other coverage that disqualifies you from contributing.
Edited to add: based on new research, a health sharing ministry is not considered "insurance" for 2020 and earlier, and so can't be a qualifying HDHP, and therefore participating in a health sharing ministry can't qualify you to make new contributions to an HSA. See below.
Whether you can contribute new money to an HSA when you are enrolled in a health sharing ministry depends on the exact benefits, premiums, deductibles, and out of pocket maximum of the health sharing ministry. The health sharing ministry should advise you on whether the plan qualifies you to make contributions to an HSA. If you already have money in an HSA, you can use it to pay your premiums and also to pay other expenses, but your eligibility to contribute new money depends on the exact plan benefits and costs and any other health coverage you might have. The exact rules are covered in IRS publication 969.
https://www.irs.gov/forms-pubs/about-publication-969
