What genre is the book The Cay? Please provide 3 examples from the book to support it.

I would say that the main genre for Theodore Taylor's book The Cay is children's literature.  Evidence for this is the fact that the book is fairly short.  The paperback version is only 156 pages long.  That's not definitive proof of the book being children's literature, but the short length makes the book much more accessible to young readers.  
The language of the book itself is accessible to young readers.  The sentence structure is not overly complex, and the vocabulary is not beyond what middle school students can handle.  Let's look at the opening of the story for an example.

Like silent, hungry sharks that swim in the darkness of the sea, the German submarines arrived in the middle of the night. 

Notice how the simile is easily visualized by young readers.  Sharks and darkness are things that kids know about at an early age, and both of those things would cause a young reader to be immediately tense.  
The fact that Phillip is eleven years old is also evidence that supports the book being children's literature.  Most authors try to create a protagonist that is similar in age to their intended readers.  Children's books tend to be about children.  Phillip isn't even a teenager yet.  
Further support for the book being a children's or young adult book is the fact that it has won awards that are specifically awarded to children's books.  For example, The Cay received the The Jane Addams Children's Book Award in 1970.  
If I had to pick a sub-genre for the book, I would say that the book is in the survival genre.  The reason for that is because the bulk of the book is about Timothy and Phillip trying to survive on a very small, deserted island during World War II.  The author does a great job of alerting readers to the very many dangers that would be in that kind of environment.  

Wondering what had caused the birds to attack me, I felt around in the sand. Soon, my hand touched a warm shell. I couldn't blame the birds very much. I'd accidentally walked into their new nesting ground.They were fighting for survival, after the storm, just as I was. 

This sequence of the book is great because I never would have thought birds could be that much of a threat.  But those birds are trying to scratch out a meager existence in the same way that Phillip is.