If I had kids I think I would encourage the exact opposite behavior. As a scientist by training I have been trained to be skeptical, question everything, and do my own research. I would hope to pass that along to my children.
I also think you've set up a straw man to knockdown when you equate being skeptical to "question[ing] the medical establishment without data". There's no shortage of data out there on and around this topic. You can search on PubMed and literally read for weeks.
I'm a skeptic, but it gets in the way of pedagogy, for example we often don't teach high school students how friction really works (according to the most up to date theories). Instead they get a few basic equations that describe a much simpler model (and which fall down if you try to make some predictions). Certainly I teach my children to be mildly skeptical, but I see no value in them questioning flossing until they're about 16 and can start to read and argue for themselves. And even then, I don't think most people are capable to read primary literature or review articles.
As to searching on pubmed and reading for weeks: that's the problem! The vast majority of people are not qualified to read the scientific literature. You need extensive training to learn how to read literature and evaluate it properly. This is especially true for medical/biology research. Learning to be proficient in that world (medical biology/public health literature) is the price of entry for being effective at changing policies. The Skeptics don't always win.
I also think you've set up a straw man to knockdown when you equate being skeptical to "question[ing] the medical establishment without data". There's no shortage of data out there on and around this topic. You can search on PubMed and literally read for weeks.