Can data included in a URL be considered secure if the connection is made over HTTPS? For example, if a user clicks a link in an email that points to https://mysite.com?mysecretstring=1234 would it be possible for an attacker to grab "mysecretstring" from the URL?
The entire HTTP request (and response) is encrypted, including the URL.
But yes, there is a way an attacker could grab the full URL: through the Referer header. If there is any external file (Javscript, CSS, etc.) which is not over HTTPS, the full URL could be sniffed in the Referer header. Same if the user click on a link in the page that leads to an HTTP (no SSL) page.
Also, DNS requests are not encrypted, so an attacker could know the user is going to mysite.com.