Why did Airi say "Whose eyes are those eyes" at the end of the first episode?
This is indeed a meta-reference, that spans across the three games in the series.

The reference starts in Chaos;Head with Takuto, he spreads the phrase and is surprise how it catches on and becomes a meme. 





One of the reasons is a serial killer that leaves behind this message at the scene of their crime.

I don't believe that there's an actual reference in Steins;Gate, but there is a reference to in the non-canonical 8-bit version:



In Robotics;Notes, "Airi" is an "automated data collection engine," so it would makes sense that she would pick up a meme such as that.