They were definitely Eldar, but the question is to think if the contdition of Calquendi was or was not hereditary. If we consider that the Calaquendi had some kind of physical "Tree-tan", it doesn't seem the sons of those will have it as well. However, if the Calaquendi were so becuase of what they have learned in Valinor, and the light of the trees is only a myth within the myth, then it would definitely be in the heritage.
The most probable thing however is that the light did have indeed an effect on the spirit, and that's why the calaquendi could move to the "world of spirits" that the Nazgul inhabited
Gandalf was discussing the confrontation with the Nazgul at the Ford of Bruinen with Frodo, Frodo asked if the shining figure he saw was Glorfindel. "Yes," Gandalf replied. "You saw him for a moment as he is on the other side: one of the mighty of the First-born. He is an Elf-lord of a house of princes." A little earlier in the same conversation, Gandalf also noted that Rivendell was home to "the Elven-wise, lords of the Eldar from beyond the furthest seas. They do not fear the Ringwraiths, for those who have dwelt in the Blessed Realm live at once in both worlds, and against both the Seen and the Unseen they have great power."
It would seem a contradiction that when Gandalf refers to these Elven Lords, he is actually refering to all Elves except their Lords: Elrond and before Gil-Galad.
My assumption is that the effect on the body and the spirit remains in th sons of the Calaquendi (although may fade) and that they are Calaquendi and High Elves as well.