Am I required to use the full zip code (including the -xxxx) for mailing packages?  Is this a rule/law?  There is one postal worker at my p.o. who is very enthusiastic about her work.  She has a lot of feedback for my office staff (who do my mailing for me) about the packages I send.

No, you are not required to use the Zip + 4.  If they say you are, ask them to show you the regulation or DMM reference that requires it. (They won't be able to.)

Sounds like I need to do the next p.o. trip myself.  I don't think my staff would be willing to confront or question this postal worker.

unless the address is a PO box, I rarely ever see the +4 given out so most people don't even know their own +4 zip.

I always use the +4, even if I have to look it up myself.  If it makes it easier for my packages to get delivered correctly, I don't mind the extra step.  

IIRC Diane is a postal employee   Although the additional 4 digits is better, I both mail and receive the old 5 digits all the time.