Take input as string because you are checking with "Done" which is string and when you take string as input through input.nextInt() it will throw an exception. I had modified you code little bit:
static void problem4()
{
int evenCount = 0;
int oddCount = 0;
int evenTotal = 0;
int oddTotal = 0;
double evenAverage;
double oddAverage;
String endKey = "DONE";
String entryVal;
int entry;
while (true)
{
System.out.print("Enter a number or 'done' to quit: ");
entryVal = entryVal.nextLine();
if (entryVal.nextLine().toUpperCase().equals(endKey))
{
break;
}
entry = Integer.parseInt(entryVal);
if (entry % 2 == 0)
{
evenCount++;
evenTotal = evenTotal + entry;
}
else
{
oddCount++;
oddTotal = oddTotal + entry;
}
}
evenAverage = (double) evenTotal / (double) evenCount;
oddAverage = (double) oddTotal / (double) oddCount;
System.out.println("Even average: " + evenAverage);
System.out.println("Odd average: " + oddAverage);
}