You can do this, if you modify some PHP code.
It is most easy to do if you have a custom theme. Then you can add a check for the browser in the layout templates, and then they affect all pages.
(However, be careful before you change the 'maintenance' layout. You probably also want to skip the text with people who have capability moodle/site:config.)
As to how to implement the check, look at the code in the quiz where the check is done, in mod/quiz/accessrule/safeexambrowser/rule.php. You can call the check_access method from anywhere.