Nope.

Who/What is Domo-kun?
Domo-kun is a small brown open-mouthed monster hatched from an egg who lives with a wise old rabbit underground. He really, really likes TV and also enjoys rocking out to Guitar Wolf. Domo-kun is the mascot of NHK's BS2 (a satellite channel; he also appears on BS1). NHK has been producing a series of Domo-kun stop-motion vignettes that exude personal charm and warmth. There is also a wide variety of Domo-kun-related merchandise available, some of which is difficult/impossible to find in the US.

"Domo" in this case appears to mean more of something (usually to stress "thank you", "excuse me", etc), implying increased politeness. "kun" is used as "Mr." or "Ms." for "younger people or colleagues". He acquired his name in the second episode, when Usajii turns on the TV for him and an announcer says "Domo konnichiwa" ("very much hello").

According to NHK, Domo-kun is recognized by 90% of Japanese schoolchildren.

The egg probably originates from the egg-shaped rings of the NHK logo (made eggy in 1995).