(1) Any egg VS chicken 

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(a) According to an article appearing in "New Scientist" magazine (www.newscientist.com) - eggs evolved roughly 340 million years ago and the first chicken evolved at around 58 thousand years ago at the earliest.

(b) According  to an article by Australian Academy of Science (www.science.org.au) - the amniotic eggs evolved more than a billion of years ago whereas chickens have been around for just 10,000 years.

In consonance with (a) and (b), it now becomes apparent that eggs came much  earlier than the chicken.

(2) Chicken egg VS chicken

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AsapSCIENCE uploaded the video "Which came first - The Chicken or the Egg" on YouTube.

     According to this video - there originated a chicken-like bird long ago, which was genetically close to the chicken. The video terms it as "proto-chicken". The first chicken originated from the egg laid and fertilized by its proto-chicken parents.

      Consequently,  the first chicken egg came before the chicken.

ARGUMENT FOR THE CHICKEN COMING FIRST

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In the problem title and the problem text, the word "chicken" comes before the word "egg".

   Therefore, the "chicken" came first followed by the "egg".