No! Students do not need a Classwork.com account to access their assignments. 

Only teachers should create Classwork.com accounts. Students access activities though the assignment links their teachers provide.

Please note that in Classwork.com Assigning means sending work to students, while Sharing means giving access to other teachers.

So remember: don't share worksheets with students -- assign them instead.

For more information, click this article: How Do I Add Students?