Hello,
Are you viewing the completion box on the course page when logged in as a student, or as a teacher? The reason I ask is because the red cross and the green checkmark only display for students and not for teachers.
Another thing to check is the activity completion settings (in the lesson settings).
If you are tracking activity completion by time spent for example, the green checkmark will only display if the student has spent enough time in the lesson.
The red cross will only display if require grade is selected AND a pass grade for the lesson has been set.
PS I am moving your discussion thread to the General help forum.