Actually...
Employer-provided health insurance is an unintended consequence of government wage controls during WWII. When an exception was made by the War Labor Board for fringe benefits, employers used health insurance as an attractor for employees instead of the higher wages they would normally have used.
Enrollment in group hospital plans tripled between 1942 and 1945. Later government rulings, including tax laws, continued to encourage employer-provided programs.
Didn't have much to do with unions, really.