She would state crazy beliefs like that homeopathy was just as legitimate as "western" medicine

Gah! That's so annoying. I'd probably just blurt out: "Homoeopathy is Western medicine, its just not very good you nincompoop."

What it felt like was undermining the precision of thought that we had trained as engineers. It felt like my brain would turn to mush by being forced to integrate these incorrect bits of knowledge.

I also know exactly how you feel I took some "liberal studies" courses too in my second year. I had the exact same feeling.