For the grammatical answer... The structure would be the same as "Hope is eternal." The verb "springs" here is acting as a linking verb, so the adjective "eternal" is correct. Other examples of similar usage: "Something smells funny." "Sometimes, love grows cold." "Humility feels unnatural to me." "George waxed poetic."

I think they meant "Hope Springs Eternally"? Eternally as an adverb; Hope springs forever.

I think "eternal" here is a 'adj".It is like the structure of "hope you happy".