A requirement to have some type of health coverage in most countries is not new, it has always been a part of a Visa or Visa Waiver travel requirement. The only difference, is now, with EES, they are having you attest to the fact, previously the border official usually did not ask (usually just asked how long you were staying, where you were going, the purpose of your visit, etc.)
Everyone should have some type of coverage, either your regular health insurance or a travel policy through a company, or credit card coverage.
There was a discussion a while back where someone linked documents showing that health coverage is not a requirement under the Schengen agreement, while true, it is a requirement in a number of Schengen countries (Keep in mind, while countries have the Schengen agreement, each country is still responsible for determining entry to their country), including, in this case, France.
I doubt you would ever be asked to provide proof, but you could. I for example believe my insurance company would reimburse me, at least to some extent, I carry my insurance card, so I answer "yes". I entered through Amsterdam last week, the question was not asked.
They also may ask if you have sufficient funds for the trip, if you have a return ticket or proof of onward travel, health status (if you appear ill or injured), and a host of other questions.