Hi Jason,
I'm using Moodle 3.4 - Students can see what groups they are assigned to by looking at their own class-level profile and the class-level profiles of others -- you can turn this feature off (preventing people from seeing their own groups as well as the groups of others on the profile page) by navigating to Site Admin - Users - Permissions - User Policies, then select "Groups" from the "Hide User Fields" box. This way, students will not see what groups they are assigned to from their profile.
As for the Participants list in the Navigation block, you may not want to go to this extreme, but you can make that clickable Participants link a non-clickable link (so, preventing students from being able to navigate to the participants page) by setting the permissions of moodle/site:viewparticipants and moodle/course:viewparticipants to n/a.