As a result, your flash file is not shown in Chrome. At the moment, Firefox and IE will still show it, but that might go away soon as well.
The solution is to serve your content from a secure http server as well.
Unfortunately, it is not cheap. My service provider charges over 70 € per year for a registered SSL certificate, and nothing else will do. However, there is a workaround: You can put your files in a your dropbox. Both links in the “public” dropbox folder and “share links” seem to come from https by default.
I quickly tested this last week, it seems to work. Also, it seems to work that these dropbox-served flash files then send their data to a php script on a regular (non https) server.
If that doesn’t work for you, please comment below.
In principle, though, dropbox-served files will probably load more slowly. So it might be a good idea to either get access to your institution’s https server, or to upgrade your own.
UPDATE 11/10/2013: it seems that is not a problem anymore? At least in a first test today, it was possible again to include non-https-served flash files in qualtrics.