IME with eels - and I have at least some (I kept a 4' Gymnothorax Undulatus for 25 years), can they get Ich? Sure. Any fish can. Being smooth skinned, however, they seem to be pretty resistant to it.
Over the years, Hannibal escaped several times. The first time I found him dry to the touch on the floor and figured he was done for, but when I touched him he opened his mouth, so I put him back in the tank and went to work. When I got home, he was swimming around the tank like nothing had happened. Although he ending up with a terrible bacterial/fungal infection, I never saw a single sign of Ich.
Despite my best efforts (eggcrate top weighed down by large chunks of rock) he got out a couple other times. Never once did I see Ich on him.
Take that for what it's worth - the experience of one aquarist with one eel, but it is my opinion that at the very least, they are more resistant than most other fishes.
hth