In the characterization of Della from "The Gift of the Magi", the phrase "she turned white for just a moment" is mainly used to show that Della is afraid of John's reaction.
The expression "turn white" means to go pale usually from shock. In this case, Della was shocked after hearing steps.
Although the phrase can be also associated to an illness, considering the context, it is very unlikely that a person will begin to get ill just by hearing someone's steps.