A lot of people seem to be having this problem, and few are getting concrete answers.
Why does Chromium say every page has insecure content?
The short answer is that the problem was being caused by the extension incognito switcher, and uninstalling it resolved things. There are posts around the internet saying that particular extension has been "hacked".
Here's how I identified the problem extension:
First, on a page that showed the warning, I looked at the console (ctrl-shift-i), and searched for the warning about insecure content. The originating page was http://www.v-tool.com/switcher/stats.js.
I searched for the file referring to this page using the following command:
After examining the matching files, it was pretty easy to find the guilty extension.