"Have health insurance" is not comparable with "is covered by universal health care" without a lot of clarification.

1. It's dependent on employment.

2. It comes with various forms of limits, out of pocket payments, and other caveats.

3. It comes with a profit motive for the insurance company to decline coverage.

etc

So fundamentally, "Am I under the threat of financial ruin if I were to get ill?" is still very much alive even with health insurance.