The links in a Showkase menu do not hardcode http:// or https:// so I would have expected that they would work fine with both protocols. (They just start with a leading slash to denote your root directory.)
Try clearing your browser's cache (or try viewing your Showkase site in a different browser) to see if this helps.
You could also try running 'System -> Setup' to refresh Showkase's knowledge of where your site is on your web server (without actually changing its location).
With regard to the "Data not saved. Unable to write xml to file page.xml" message, default permissions of 644 should be fine for your 'page.xml' files (inside your Showkase page folders). I've seen some web servers configured such that 777 permissions are seen as a security issue and using 777 can cause unexpected problems.
If none of the suggestions above help, your could perhaps recreate your Showkase site from scratch (in a new subdirectory) and import all the pages from your existing site (via 'Site -> Import') to prevent you from having to recreate your pages. You could then either move your site (following the instructions here) or just make your Showkase home page your site's home page (following the instructions here).
(Creating a new site would be a last resort and should hopefully not be necessary but I thought I should note it as a possibility. If you do this, I would recommend that you make a complete back up of your existing site first in case anything goes wrong and you need to reinstate the files at a later date.)