What do you think happened at the end of the story? What does he mean when Mr. White beg his wife not to let "it" into the house? What is he afraid of? Who or what was outside of the house?

I think that we can infer that both Mr. and Mrs. White think their son is at their door in zombie form,

A cold wind rushed up the staircase, and a long loud wail of disappointment and misery from his wife gave him courage to run down to her side, and then to the gate beyond. The street lamp flickering opposite shone on a quiet and deserted road.

It is not clear what they will do about the knocking at the door. It is possible that Mr. White might use his final wish to send his son back to the grave. Mr. White cares for his wife so whatever happens depends on how Mrs. White reacts to the noises outside.