We set up a Google Apps Script-based Intranet in our organization, which benefits from Google Apps for Education. We\'ve been using for months:
Same thing has been happening to us. We have a G Suite for Nonprofit account (formerly Business) and have published webapps for several years via this account without the warning showing up. Users began telling me yesterday July 6 2017 that the warning now appears across all our webapps.
I just created a test webapp where "Execute the app as" is set to an admin of our account and "Who has access to the app" is set to "Anyone, even anonymous". As you can see https://script.google.com/macros/s/AKfycbxrKgPol_-eEskKzqRzN4WAMj-NiPm7iFavjZu-_vWtTJPyAEkK/exec does display the warning.
Something I just noticed, the generated webapp script's url now starts with script.google.com/macros/s/ whereas before it would have included our organization name (e.g. script.google.com/a/macros/rchsks.org/s/). So, the url will change if I go back to redeploy one of our existing webapps -- that will screw up several things, the least of which are bit.ly links pointing to the old url.
Not having these warnings appear was one of the benefits I read about when we considered signing up for what was then called a Google Apps account. I'm an admin of our account and received several G Suite alert emails on July 6 2017 about problems occuring with Google Drive, Calendar, Hangouts, etc. Maybe those are connected?