Binary Heap Order Fixing in Java
if (this.heapType == 0) {



and 


    } else if (this.heapType == 1) {



Rather than using magic numbers, it is typical to use Enum values for this in Java.  

    if (heapType == HeapType.MIN) {


and 

    } else if (heapType == HeapType.MAX) {


or in the latter case, just 

    } else {


Presumably there are only two types of heap.  

You don't have to say this.heapType.  It's enough to just say heapType if there is no conflict.  You can add the this. if you find it makes things clearer, but it's not functionally necessary.  You only need to specify this. if there is another variable with the same name as the object field.  

Another way to do this would be with an interface and separate implementations for Max and Min heaps.  That may not be better though.  It's hard to comment with so little context.  It would save the if/else here but might cause problems elsewhere.  

In general, I can't comment on how this method works with other methods like the caller or swapValues, as those definitions are not included.  Should you work directly with the value rather than the node?  Should this be integrated into the caller's logic?  I can't really tell.