You need to check with the place where you want to obtain insurance what proofs it wants to sell you the insurance. There is no statutory approved proofs if you are not legally married or in a state registered domestic relationship.
Only certain persons are eligible for a state registered domestic relationship. More information is at Chapter 26.60 RCW (STATE REGISTERED DOMESTIC PARTNERSHIPS).
WA does not allow the formation within WA of a common law marriage between two persons regardless of how long two persons live together.
While there is the doctrine of committed intimate relationship that may apply to your situation, that doctrine has no application to health insurance. Under the doctrine, the persons have property rights in properties that would have been community properties had they been married with each other.
Some employers allow their employees to enroll specific persons in the employees' health insurance even though those persons are not the employees' spouses, state registered domestic partners, or dependents.
You need to check with the place where you want to obtain health insurance to find out what proofs it wants. In some places, a statement signed under oath (such as through a public notary) may be all the proof the employer/insurer wants.
It would seem to me that the insurance company or Representative would have a list of the things that would qualify for this purpose. I'd suggest checking with them. Good Luck!
THIS IS INTENDED TO PROVIDE GENERAL LEGAL INFORMATION AND CONSULTATION WITH A LEGAL PROFESSIONAL IS ALWAYS RECOMMENDED SO A FULL EVALUATION OF YOUR SPECIFIC SITUATION CAN BE PROPERLY ASSESSED. PROVIDING THIS LEGAL INFORMATION DOES NOT CREATE A LEGAL RELATIONSHIP BETWEEN THE PARTIES.