That's not a problem, your students can join your closed group having an email adress or not.
You can send them the link to join the group, as you can see in the following screenshot:
and you students will only need to create a username and password for themselves and accept the invitation for your group.
You can also send them the code that appears on the same page (on the screenshot is hidden) and they can go to the student signup page and type down the code and create an username and password as before.
Hope that solved your question and enjoy the benefits of the closed groups!