I wrote this question after searching “health insurance” on the Library of Congress’s, online Chronicling America collection, and finding an early health insurance company. When I Googled to ask what the first health insurance company was, I find all the answers to be incorrect. According to the U.S. Census Bureau’s website, “The Massachusetts Health Insurance Company of Boston was organized this month [April] in 1847". According to the Insurance Information Institute, in 1850 the Franklin Health Assurance Company of Massachusetts offered the first accident and health insurance.. The U.S. Bureau of Labor Statistics seems to endorse the Franklin Health Insurance as the first health insurance company by posting a paper to their page that says, “For a 15-cent premium, the policy would pay the bearer $200 in the case of bodily injury as a result of an accident by railway or steamboat. If the accident resulted in total disability, the policy would pay $400..
Both answers are incorrect. The reason they are incorrect is because at least one health insurance company predates these companies.
Also, thanks to a comment, I'll explain what I mean by "health insurance" and "company." By health insurance company, I simply mean a private organization founded with the purpose to provide health insurance to others. This would exclude government attempts at insurance, like that mentioned in Zach Lipton's comment. This would also exclude, arbitrarily I agree, mutual benefit societies, like that of the Odd Fellows (mentioned in my answer below), because they provide much more than simply health insurance. For example, unlike health insurance companies, the Odd Fellows provide education services for orphaned children, and care for widows.
"Health insurance", as referred to here, means insurance paid during times of illness or injury. Now, "insurance" is different from the modern use. In modern parlance, health insurance simply covers the cost of medical expenses; however, back in the mid-19th century, medical costs were minimal. This means the primary cost of illness or injury back then was lost wages from inability to work. Therefore, "insuring health" back then meant that the insurance covered the cost of lost wages by paying a weekly check during the "calamity".
In some places, such as a JSTOR article I can no longer find, or Edwin J. Faulkner's 1940 book "Accident and Health Insurance" (I can't find it online, I got it through my library), they differentiate between "accident", "sickness", and "health" insurance. The first being money paid when the insured incurs an accident; the second when sickness is incurred; with the third being money paid for healthcare. These are later classifications. At the time, "health insurance" referred to both sickness and accident insurance, as my answer will make clear.
If anyone thinks necessary, I could add the history that predated private insurance, but this is kind of long as it is.