I had the same problem with version 6.6.

I resolved it by following a few steps:

1.  Deleting all of the .browser files in the App_Browsers folder as they were no longer valid for IE11, and installing the latest service patches on the web server to get new default browser files.

2.  Delete all version 6.6 cutesoft files and dlls, and then install the new  version 6.7 files and dlls by downloading them from this site.

3.  Ensure this is still in web.config appSettings: 

< add key="CuteEditorIgnore304" value="True" />

HOWEVER there are still a few javascript errors/warnings/bugs that are still unresolved; but at least most functionality works:, especially if you disable javascript warnings.  Errors listed here:

http://cutesoft.net/forums/thread/78301.aspx