I have a similar problem: when adding an url of an image, say from Wikimedia, the image doesn't show up.
And we have SSL.
The Silverlight runtime does not support mixed-mode content (e.g. mixing HTTP and HTTPS content), so Explorer Online cant show images accessed over HTTPS - could that be your issue? (doesn't really explain why they worked initially, unless your server was serving the images over HTTP previously, and then that changed).
(Let me know if you're using Explorer Online as part of the beta program for ArcGIS for Organizations, as the situation may be slightly different depending on what options your organisation has set).
I simply removed the 's' from the https and reloaded the Topographic basemap and for most images this solution seems the way...
But I don't know why I was in HTTPS mode...
what are therefore the advantages of working in HTTPS mode when just creating an online map in Explorer Online?