Eidolon And Improved Natural Armor

Is there anything preventing an eidolon from taking the improved natural armor feat in the Bestiary?

Nope, You could, if you wanted to, take Improved Natural Armor as many times as you wanted as you qualify for the feat

Assuming you put at least 1 point into Natural Armour at the time of taking the feat, there is no reason why not. (Beyond using a Feat for something you can improve with Evoultion points)

Assuming you put at least 1 point into Natural Armour at the time of taking the feat, there is no reason why not. (Beyond using a Feat for something you can improve with Evoultion points)

All eidolon base forms have natural armor (at least from the APG, I believe the aquatic does as well). Prereq is an innate ability for them. The evolution has a level limit 1st and every 5th summoner level you can choose it again, the feat doesn't.