It depends what you are trying to s

It depends what you are trying to say. They are all viable sentences, in the right context.

There is some egg left, I not sure if it supposed to be scrambled egg, or fried egg, and it is cold, but you should eat something.
Why don't you eat some egg?
That is an offer of egg (cooked together to make a dish)

I can cook you an egg. A perfect boiled egg, three minutes, with soldiers.
Why don't you eat an egg?
That could be an offer as well. (An offer of an egg. Distinguishable as an egg)

I am an ovo-vegetarian, which means I don't eat any meat, or any fish, but I do eat egg.
Why don't you eat any egg?
That is only a question as to their eating habits.

Sorry!

If the question is about articles, it means 'an egg'.
If the question is about uncountable words for food, it is 'some' egg.
If it is about uncountable label for a foodstuff, it could be 'any egg'.
Purely my opinion!