I did a quick test, making my local server feeding HTTPS pages. My Odyssey V.1 was able to download EPUB files from it. Note that the test page was a plain HTML page, with links as <a>'s, and my web server was administering the correct MIME type for EPUB.
Then I tried to download a file over HTTPS from my Wuala account, and the web page opens, but it is unusable, probably because the integrated browser cannot cope with the JS'es which infest that page. (Note: I forget to try DropBox mobile, but even that page has plenty of JQuery and the like.)
So, I guess it all boils down to the rendering capabilities of the browser, more than the HTTPS thing.
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