aoneil,
First, create a data selection for your address block that contains all names, addresses, companies... Then, double-click on it, go to "Lines", and select "Skip completely empty lines". Then, go to Snapping Points, and check set one of the snapping points. The lower-left one should be good.
Then, create a barcode object, and select your zip code as the data. Choose POSTNET as the barcode type in the Barcode Options. Then, go to Snapping Points, and snap the barcode to a previous object. Use the upper-left box.
Now, whenever the company info is missing, it won't be displayed, and the barcode should move up. Try it out, let me know if it works!
Regards,
Rapha