Yes, you can use Quill without creating student accounts or saving student data. Every Quill activity is publicly accessible from https://www.quill.org/activities/.
You can share a link with your students, and they can complete activities without creating an account. However, since it is an anonymous session, the results will not be saved and you cannot view them on your teacher dashboard.
For each activity would like to use with your students, copy and paste the "view activity" link to save a copy of the link. You can then share the link through a service like Google Classroom or by writing the URL on the board.
Go to the Quill Activities page and find the activity you would like to use. https://www.quill.org/activities/
When you find an activity you would like to use, right click on the "view activity" link and then click on the "copy link address"
Paste the link in a location your students can access.
Create A Short Link
If you would like to write the link on the board for your students, you can make a short link to make it easier for your students to re-type the link.
Go to bit.ly
Paste the link
Edit the URL to make it easy to re-write.