Up through the fist half of the 20th Century, medical expenses were generally paid individually, or through mutual-aid groups such as churches or fraternal societies. This began to change when the government instituted wage freezes. In that environment the only way for an employer to attract superior talent was to offer non-salary benefits, and the tax advantages of health care made that an attractive one.
Thus, American employers began to offer health insurance as a way to better compensate their workers when the government was trying to interfere with the labor market. As the practice became more common, it increasingly became a point that job hunters would look for. And the rest is history.
If you think that the results of this meddling are undesirable, then you might think twice about giving the same meddlers the authority to muck things up further.
Edit: here's a bit on FDR's wage freeze and its effect on health insurance: http://faculty.smu.edu/tmayo/health%20care%20timeline.htm