Yes, teachers have the ability to select which particular students should be assigned a written assignment. Creating multiple versions of assignments for one story is a beneficial feature to differentiate supports based on students’ needs.
After you have created an assignment, click on a class name to reveal your student roster. You may then select who you would like to include and remove from an assignment.
Here's a video of the experience:
This video covers the written assignment experience, including how to assign to specific students: