@heath_freeland, does no autosaving work on quizzes in your course or just on this single quiz?
I saw one post that mentioned that auto-saving is triggered by moving onto the next question. That's something you could test with Student View. When I just tested in Student View, I noticed that closing the tab/window without navigating away from the quiz (within the course) caused the answer to not be saved.
Are the questions on the quiz auto-graded or manually graded? If they are auto-graded, I might suggest having the students submit each time rather than relying on the auto-save feature. If they are manually graded that would work as well, but I am not sure of the size of your class, so I do not know if it is feasible to go in and grade each check-in.