I just wanted to update what I wrote April 9: The problem turned out to be a malicious attack on the site. Our host (GoDaddy) confirmed this--albeit slowly--and recommended a password change and then a complete site rebuild from scratch. This seems to have worked. I'm still patrolling the site regularly for unexpected changes in .html, .js, and .aspx pages. Whatever program the hacker used even attacked some legacy pages that I had left on the site from laziness, a kind of *.html.bak arrangement that I had originally used in case my lousy code editing skills didn't work. Yes, even those pages were hit.
A couple users did complain of machine stalls, and one confirmed that her anti-virus program (Kaspersky) stopped a trojan in real time. We have recommended all users between March 27 and April 9 scan their machines thoroughly.
The extent of the attack was breathtaking. GoDaddy does not allow a DNN site in the overall root folder, so we use a redirect page--that was hit. The siteofline.html page was hit. Our .aspx error page was hit. Even the 404 error notification page was hit. Shutting down the site in order to protect customers was a major exercise in thinking about how many error pages our site actually has.
Rebuilding was not a snap, but I have been faithful in making DNN backups, and that eased the process considerably. Because of form issues, part of our site is actually outside the DNN portal, so I now make sure to back up everything on the website.
Lesson learned: it is a rough world out there.
Ken