Translation:We are making use of the free health insurance.
Well I think that in German it helps a lot to cut a word to remember it and get its full meaning ! (Eg. Regenschirm, Krankenhaus) Besides, maybe "disease insurance" sounds awkward in English, but it is also how we would say it in French (assurance maladie) so it is not that absurd ;-)
They used to call it "assurance maladie" in Ontario (Canada), but changed it some time ago to assurance santé. (In English, I think it has always been "health insurance".) In Québec, they still call it "assurance maladie". I guess it's whether you consider it insurance against something, or for something.
I think "medical care" should be accepted. Health coverage isn't "insurance" in the same sense as house or car insurance. Even if the term "insurance" is used in German, that doesn't make it so. Whether or not you agree with my take on this, though, there are synonyms for "health insurance" in English, such as "health coverage", which at present (Dec. 2020) is not accepted by Duolingo.
"No cost" translates the same as 'kostenlose.' Thus, DuoLingo got the sentence correct in German.
However, the supposed 'correct' answer in English is wrong.
It should actually be: "We use the no cost health insurance." An alternative is: "We use the no-cost health insurance."
Whoever is paying for someone else's coverage, it is certainly not "free" to those payers. But it is at 'no cost' to the individual who takes advantage of it.
In English, we can say "health insurance", or if the context is clear, simply "insurance". The reason we need context is because there are other types of insurance, which I'm sure you know.
Krankenversicherung is simply a compound noun, with the second half being the word "insurance": "Versicherung"; thus if the context is clear, you can simply say insurance in German.
"We are using the free health insurance" is good English. I don't understand what your alternative suggestion is exactly. Perhaps you mean "make use of"? "We are making use of the free health insurance" is also good English. Both sentences mean the same thing and should be equally acceptable. The same is true of "We use the free health insurance," which is the most literal translation of the German.