"When I hear someone say, "Here, here" as a gesture of agreement to the comments of another, I don't know which to use."
I would lean toward "hear, hear", as in having people hear what is being said because of being in agreement. Or possibly a shortened form of "hear ye, hear ye."
I will admit to succumbing to the use "their" when his/her is more appropriate.
I am not following the "to eggs" business at all.
But then I could probably use a refresher on who has the right-of-way at four way stops, too.
"Hear! Hear!" is correct as in "Hear him!"- listen to him.