A Nightmare has the ability: Ethereal Stride:
The nightmare and up to three willing creatures within 5 feet of it magically enter the Ethereal Plane from the Material Plane, or vice versa.
The party's wizard claims he should be able to use counterspell to stop it since it is magical.
Can counterspell stop a Nightmare's Ethereal Stride ability?
If it can stop it, at what level should it be considered to have been cast?
And as a bonus question, would the same answer apply to a night hag's Etherealness action which requires a heartstone to use (and therefor in my mind is closer to being a spell)?