If it helps, when looking at eels in the
LFS and at Petco, they all have odd breathing habbits like that. It almost looks like they are filling lungs instead of gills.
They sit there for a couple seconds, then take a nice big inhale and small sacks seem to inflate behind their heads. Then they let it out and sit for another couple seconds.
I do not know if this is healthy behavior or not, but i have only seen a handful of eels, and all of them breathe like this.