Give four examples that prove that John's society was a very superstitious society.

Paragraph one of "By the Waters of Babylon" gives a lot of evidence to support the statement that John's society is a superstitious society.  The paragraph begins by stating that it is "forbidden to go east."  There is no explanation about the rule, but the entire society follows it.  Perhaps that doesn't scream superstition, but because the sentence is immediately followed by talk about the "Dead Places" and the fact that only priests and sons of priests can go there, the paragraph heavily hints at superstitious beliefs in otherworldly powers.  

Also in that paragraph is the information about the "Place of the Gods."  The information tells readers that John's society believes in beings of higher power who must be feared and respected.  That's superstitious.  Add to that the paragraph also specifically mentions spirits and demons that live in the Place of the Gods.  

It is there that spirits live, and demons—it is there that there are the ashes of the Great Burning.

Also, the fact that John's society has priests tells readers that the society is superstitious.  The people need priests in order to intercede on their behalf to the Gods.  

Once John begins his priestly training, he tells the reader that he learned how to do basic first aid techniques; however, the members of his society believe his skills are due to magic, chants, and spells.  It's mystical and superstitious stuff, not science.  

I would say the last major piece of evidence is the importance that John's society places on dreams and the meaning of dreams.  

"My son," he said. "Once I had young dreams. If your dreams do not eat you up, you may be a great priest. If they eat you, you are still my son. Now go on your journey."

The only reason John is allowed to go east is because his father believes that a dream has called him to make a spiritual journey of sorts.