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Accessing Squirrel mail through https seems to work fine atleast on a "shared" installation.
Don´t know how it would work if you have your own installation but I would imagine it would be the same result.
You should not need to do anything to use Roundcube so please submit a ticket with the helpdesk and the techs will look into it. Link on top of page and in my signature.
I would log in to the file manager (or through ftp) and see if anything suspicious has been uploaded and check all email accounts for things that may have gotten stuck there.
After that I would ask the helpdesk to look into it.