Let me give you the IT management perspective on this:
You may have a better case if you complain about your department's data protection practices - that's assuming that the shared mailbox used is the only one at your disposal.
A shared mailbox is, by definition, used by several people - and there may always be legitimate reasons why people would check other stuff than strictly the inbox. This even includes deleted items.
Three of the teams I manage have shared mailboxes (though everybody has a personal email account as well). They're under strict instructions only to use them for non-personal, whole team responsibility correspondence and must even sign mail sent via the account with "A team" rather than "name".
As a manager, I'd expect sick notes to be sent from my employee's personal mailbox - along with other stuff that's not meant for everybody, e.g. complaints, requests for 1:1s, concerns with business practices etc. As a matter of fact, I'd consider use of a shared mailbox for such matters a case of possible misconduct (depending on content) and might check with HR.
But that's because all my employees have personal mail accounts in addition to shared ones.
Think of it like so: shared mailbox = holding your conversation between the desks at your open plan office. Personal mailbox = meeting room/boss' office/etc.
Obviously, this depends on there being a clear IT policy in place and employees having access to alternative means of communicating with the boss.
If your workplace doesn't ensure that's the case, from an IT management perspective, you may have reasonable grounds to claim that your workplace is in breach of data protection laws. OTOH, if you do have other means of communication, I might question your judgment in using a shared mailbox.
From a pure IT perspective, your problem may be that your colleagues may have legitimate reasons to dog through email and may hence have found your note regardless of whether or not they were deliberately digging for it. Which, again from an IT perspective, makes your case against your employer instead of your colleagues a much better one, providing you exercises sound judgment in using the mailbox in question.
HTH.
Also: sorry this happened to you - it does sound like a crap situation. 