Hi, Lucy--
Dixie cups are a brand of throwaway small paper cups,
designed to be pulled out of a dispenser next to a water
cooler or drinking fountain. I believe they were touted as
being more hygienic than regular glasses, or the kind of
fountain where the person in front of you might have put
his germy mouth right on the spicket. They became VERY
popular, even in private homes, during the polio scare
of the '50's.
I think the reason Anna used that term is that she lives in
"Dixieland". (Technically, I do too, but Kentucky was neutral during the American Civil War, and is not the "Deep South" geographically.)