"you should know that I still don't trust him. Now he says that Neal was authorized to give his idol but it doesn't prove a thing as far as I'm concerned."

I believe the only other person Medically Evacuated while holding a Hidden Immunity Idol was Colton in One World. Colton asks Jeff if he can give it to someone, and Jeff tells him that it’s his decision. Colton tells his tribe that he’s going to keep the idol, and thanks Sabrina for the souvenir.

So there is precedent for Neal to be able to pass his idol to someone else before leaving as a Medical Evacuation.

Also in Neal's exit interviews he says he was allowed to pass the idol on. For instance:

Survivor: Neal reveals why he did not give Aubry his idol
http://www.ew.com/article/2016/03/31/survivor-neal-insidetv-podcast-norman-reedus-rachel-bloom

On his decision to not give Aubry the idol:

“That’s a decision that I will probably go back and question for the rest of my life. But in the moment it was the one physical thing from the game that I love very much that I could hold onto. And the night before when I let her know that I had the idol she also made it clear that if she had to, she would have no problems slitting my throat. And in the moment, knowing there was such a good chance that I was going home the next day because of an evacuation, that wasn’t funny to me.

“So it was kind of a combination of factors. I thought on one hand, I gave her life by my life being extinguished in the game and it got her at least three days further. And sometimes the threat of an idol can be as powerful, if not more powerful, than an idol itself. What you didn’t see was me giving her a jacket, and people didn’t know what was in that jacket.”