I have figured how to successfully search for strings with special characters with the exception of open curly bracket characters '{'.
As an example, this query works:
issueFunction in issueFieldExactMatch("project = myproject)", Summary, "search + - & | ! ( ) } [ ] ^ ~ * ? \\\ : test")
Every special character is in the Jira I'm searching on and it gets properly returned in the result set.
If I add the open curly bracket, no escaping I can come up with makes it work (double backslash, unicode, etc.