It means that Miss Ginny thinks she is so sick she may die.
Missy Ginny thinks that she is "hangin' on the drop edge of yonder" and that she needs a doctor. These lines tell us that the bad ham she ate has made her very sick. The "drop edge of yonder" is another way of saying that she is hanging on for dear life. This shows that she feel she is about to die since she is so sick. Shut is referring to her life ending not a door.
