There is an old shared connection that I had setup to a PC to transfer files from the PC to the Mac and I cannot get rid of it.
Here is a shot of the message when I drag to the trash; control click give no trash option.
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Try disabling / re-enabling Bonjour:
Uncheck Bonjour computers, wait a few seconds, and re-check it. That should clear out the shared computer.
Devices shown in Network are reported by the network, not by your machine, hence they can't be deleted. Try restarting your router to clear any persistent assignments.