The Eggnet was a global communications network that was used by the Eggman Empire.
Usage by Neo Metal Sonic
While enacting his takeover of Angel Island, Neo Metal Sonic posted a major update regarding his plans on the Eggnet which he did not bother to hide. Sonic the Hedgehog and Silver the Hedgehog later managed to infiltrate one of Eggman's bases and access the Eggnet. After meeting Whisper the Wolf and her Wisp companions as well as defeating the Badniks residing there, Whisper almost destroyed a console that the Eggnet was on. Before she could, Sonic stopped her so that they could first get any vital intel off the computer. That was when they discovered Neo Metal Sonic's plans, which prompted them to mobilize the Resistance. (StH: #8)
The Metal Virus
While working on his plans for his Metal Virus, Dr. Eggman would put his files out on the Eggnet. Some time after, Sonic and Amy Rose managed to successfully log onto the Eggnet from the Echo Mine's computer. Their plan was to find out what they could about Eggman's latest plan. Just as Amy found files on an ARK-like space station and talk of "payload distribution" however, Rough and Tumble arrived and destroyed the computer with a tank. (StH: #15)