Your condition in the for loop used a > when it should be <
for (int count = inputMonth; count < inputMonth2; count++) {
    middle = middle + Days[count];
}
Your code contains a multitude of logical errors (for example: what happens when 2 dates in the same month are used), below is a working version:
public static void main(String[] args) {
    String monthStr;
    String dayStr;
    int inputMonth;
    int inputDay;
    String monthStr2;
    String dayStr2;
    int inputMonth2;
    int inputDay2;
    int code;
    int code2;
    int start;
    int middle = 0;
    int end;
    int sum;
    String[] Months = { "January", "February", "March", "April", "May",
            "June", "July", "August", "September", "October", "November",
            "December" };
    int[] Days = { 31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31 };
    monthStr = JOptionPane.showInputDialog(null, "Enter First Month: ");
    inputMonth = Integer.parseInt(monthStr);
    dayStr = JOptionPane.showInputDialog(null, "Enter First Day: ");
    inputDay = Integer.parseInt(dayStr);
    monthStr2 = JOptionPane.showInputDialog(null, "Enter Second Month: ");
    inputMonth2 = Integer.parseInt(monthStr2);
    dayStr2 = JOptionPane.showInputDialog(null, "Enter Second Day: ");
    inputDay2 = Integer.parseInt(dayStr2);
    if (inputMonth == inputMonth2) {
        // Same months
        sum = inputDay2 - inputDay;
    } else {
        // Different months
        start = Days[inputMonth - 1] - inputDay;
        end = inputDay2;
        middle = 0;
        // Start at inputMonth+1, start already included the days remaining
        // in inputMonth
        for (int count = inputMonth + 1; count < inputMonth2; count++) {
            middle = middle + Days[count - 1];
        }
        sum = start + middle + end;
    }
    JOptionPane.showMessageDialog(null, "DAYS COMPUTATION PROGRAM"
            + "\nFirst Date: " + Months[inputMonth - 1] + " " + inputDay
            + "\nSecond Date: " + Months[inputMonth2 - 1] + " " + inputDay2
            + "\nNumber of Days Elapsed: " + sum + " Days");
}