I think I found the answer and the answer is NO!
Hollister
It sounds like you've connected your PCs as a Network. There are no IP addresses used in a one-to-one serial connection. It's just three wires -- read, write and common, with the read and write crossed over.
GPSout and FliteStar connect by serial cable. there is no GPS input to FliteStar other than on a serial port as far as I know. Did you check?
The README file I supply clearly says "connect your flying PC to your moving map PC with a serial "null modem" cable (ie send and receive crossed over), and configure your map program accordingly.". I do not know of any NMEA standard for GPS connection via Network, whether Ethernet or USB or any other means. I think it only works with a simple dedicated COM port (serial) connection.
If you know of a moving map application which can receive standard GPS input by any other means please let me have full details, and also the GPS devices which can provide such output.
Have you checked the FliteStar documentation to see how it wants the connection? My FliteMap only allows COM ports to be selected, with selectable line speed. You can run it in test mode to see what is arriving and