"everything" is a specific concept. :) If you say "everything" you know exactly what you are talking about (i.e. all things). It is also a direct object, so it is the perfect example of something that gets the accusative.
As knowbody seems to answer your question, I will try to do it.
That her şey will turn to her şeyi is not depending on the words around. When her şey is the direct definite object of a sentence, then the accusative is used. Then it must be her şeyi,