When there was only 1 golden egg around (that is, after the first Easter egg hunt), it was giftable. Tom won it then gifted to Jennifer.
The second Easter egg hunt, the second golden egg was NOT giftable. And my understanding is, the original egg became ungiftable as well.
Because the second golden egg was taken away, there is only 1 golden egg right now, and it belongs to Jennifer. She's a pretty devout collectibles collector and also, has became inactive. Even if it is still giftable, I don't think we'll ever see it leave her possession.