Is a egg penguin a proper noun?

No, the noun 'egg' is a common noun, a Word for any egg; the noun 'penguin' is a common noun, a Word for any penguin; the term 'penguin egg' is a common noun a Word for any penguin egg.

A proper noun is the name of a person, place, thing, or a title; for example:

• Egg Harbor, WI or Penguin Island, Shoalwater, Western Australia
• The Egg House, Simi Valley, CA or Penguin Hotel, Miami Beach, FL
• 'Patty the Penguin Lays an Egg', iphone and ipad app