Вариант с временным массивом.

int newArrayCount = 0;
        int arrayCount = 0;
        while (arrayCount< array.length){
            if (array[arrayCount]==null){
                    arrayCount++;
            } else {
                newArray[newArrayCount] = array[arrayCount];
                newArrayCount++;
                arrayCount++;
            }
        }
        for (int a = 0;a<array.length;a++){
            array[a] = newArray[a];
        }

for (int i = 0; i < array.length; i++) {
            if (array[i] != null)
                temp[i] = array[i];
            else if (array[i] == null)
                temp[array.length -1 -i] = null;
        }
        for (int i = 0; i < array.length; i++) {
            array[i] = temp[i];