I know it's wonderful that you keep improving the site, but I worry that I'm about to be 'improved' right out of my ability to access it. If you switch to a https site, I won't be able to access it on my home computer, which is ancient and cranky and won't go to secure sites. I'll have to read my stories at the library, which is a pain.
I admit I am not very experienced with Internet programming, but it might help to try switching / upgrading your browers either to Firefox or Google Chrome. If that problem persists, please check within the browers the security levels, try setting "default level". Additionally, you can manually allow single websites to adding them to "trusted websites", or whatever that category is called within your choosen browser. Check if date and time are correct, some certificates check these as well. Lastly, it might help to look under Internet Properties, Content, Clear SSL state.If all this doesn't help, you can search for Windows updates, then there might be a dll missing, or some other reason...