I believe I must have one of the newer network cards.
Page 11 of 
https://www.chem.agilent.com/Library/qu ... etwork.pdf Agilent LAN network guide describes exactly how I'm able to set the IP address (holding down the 'load' key during startup, manually programming IP address, etc.).  BootP won't work with the LAN card I'm using.  This is an isolated configuration by the way.    
Oh, and AICMM I did try your suggestions for using the default IP address on the PC, but that didn't work.  Besides, I can clearly see the IP address of the GC when I hold down the LOAD button while restarting the GC.  With the testing I've done, I've found the following things:
1) The NIC card of the PC receives a signal when plugged into the company's network.  Therefore, it works.  
2)  The NIC card of the PC does not receive a signal when plugged into the network card of the GC.  This was attempted using a second PC which also produced no signal.  
3)  The crossover cable was replaced, but it had no effect.  Therefore, it's not likely to be a cable issue.
4)  The IP address of the GC (10.1.1.101) corresponds to the IP address of the PC Network Card (10.1.1.100), differing only by one number.  This has no effect.
At this point, I don't know what other conclusion to draw other than a hardware issue with the GC that is causing no signal.