Why is 鸡 required here? The type of eggs aren't specified in the English statement...
In English, eggs are chicken eggs unless otherwise specified. In Chinese, the name of the animal that the egg comes from is always included.
You can simply put 鮮
Yes, that is the traditional character of the form 鲜. In traditional characters, the sentence is 這些雞蛋不鮮。
Hm... not to the nitpicking me. Only for describing the taste I leave out 新.
Lots of fruit stores around here have 鲜 in their names. I really don't think "new" (新) has to accompany it in this sentence.
Can you not say 这些鸡蛋不很新鲜？Does 很 not work in the negative?
I believe you can say that, but the meaning is a little different. In that case 很 distinctly means "very" (it's not a filler word.)
这些鸡蛋不新鲜。=These eggs are not fresh.
这些鸡蛋不很新鲜。=These eggs are not very fresh. (So maybe the eggs are somewhat fresh?)
In the negative, 不 replaces 很. In the positive, 很 connects the noun and adjective and can mean "is" or "very."
我很高兴 - I am happy
我不高兴 - I am not happy