How do I display a string from an array from user input?

To prompt the user to enter a county, and then display it (without a switch) is simple enough. You could use something like,

String[] county = { "Carter", "Cocke", "Washington", "Greene",
        "Hawkins", "Johnson", "Sullivan", "Unicoi" };
Scanner scan = new Scanner(System.in);
for (int i = 0; i < county.length; i++) {
    System.out.printf("%d %s%n", i + 1, county[i]);
}
System.out.println("Please select a county from above: ");
int choice = scan.nextInt();
if (choice > 0 && choice <= county.length) {
    System.out.println("You selected: " + county[choice - 1]);
} else {
    System.out.println("Not a valid choice: " + choice);
}

As for testing if a String array contains a particular String you could write a utility function using the for-each loop like

public static boolean contains(String[] arr, String val) {
    if (arr != null) {
        for (String str : arr) {
            if (str.equals(val)) {
                return true;
            }
        }
    }
    return false;
}

i was working on the same thing, use ArrayList. this was my code

import java.util.*;
 public class Practice{
    public static void main(String[] args){
       ArrayList<String> mylist = new ArrayList<>();
        mylist.add("Maisam Bokhari");
        mylist.add("Fawwad Ahmed");
        mylist.add("Ali Asim");
        mylist.add("Maheen Hanif");
        mylist.add("Rimsha Imtiaz");
        mylist.add("Mugheer Mughal");
        mylist.add("Maaz Hussain");
        mylist.add("Asad Shahzada");
        mylist.add("Junaid Khan");
        System.out.println("Name of the student: "+mylist);
      }
  }

now if u want a specific name from the list put this in system.out.println
System.out.println("Name of the student: "+mylist.get(1));

now the trick is to let the user enter the number in get( )

for this i made this program, here is the code

first make a scanner

    Scanner myScan = new Scanner(System.in);
    int a = myScan.nextInt();


    System.out.println("Name of the student: "+mylist.get(a));

now it will only print that name depending on what number user have entered !!

I didn't understand well your questions, but I am going to answer on what I have understand.

In case you want to print the value of a given index by the user here is the solution: Try an i with correct (existing) index and another one which does not exist e.g i=9

public class app
{
   public static void main(String[] args)
   {
      String [] county ={"Carter","Cocke","Washington","Greene","Hawkins","Johnson","Sullivan","Unicoi"};
      int i = 10;  //i is the user input, you should do that using a BufferedReader or Scanner.
      try
      {
         System.out.println(county[i-1]);
      }
      catch(IndexOutOfBoundsException e)
      {
         System.out.println("This index doesn't exist");
      }
   }
}

In case you want to check if a given word exist you can do it that way: Again try a string which exist and one which does not exist.

public class app
{
   public static void main(String[] args)
   {
      String [] county = {"Carter","Cocke","Washington","Greene","Hawkins","Johnson","Sullivan","Unicoi"};
      String word = "Cocke";  //word is the user input, you should do that using a BufferedReader or Scanner.
      boolean found = false;
      for(int i=0; i<=7; ++i)
      {
         if(word == county[i])
         {
            found = true;
            break;
         }
      }

      if(found == true)
      {
         System.out.println(word + " is in the array.");
      }
      else
      {
         System.out.println(word + " is not in the array.");
      }
   }
}