Can Buddhism lead to eternal happiness? Does Buddhism propose the possibility of any state of consciousness which feels better than neutral?

Many buddhist teachings I have heard so far (mostly by Alan Watts) seem to point to an "ideal" state of consciousness which feels neutral or maybe even not at all, as there is no conscious. 

Does Buddhism claim to have some path to "eternal happiness" and more than "non-suffering"?
That's a very common misunderstanding of what Buddhism is after.  Western authors (especially some of the early dabblers in Buddhism like Watts) had a very difficult time expressing Buddhist ideas in the parlance with which they and their audience would be familiar.  Consistently, concepts like "voidness" (e.g. shunyata) get mistaken for ideas like nihilism, inertness, or Stoicism.  You'd half expect the Buddhist ideal to more resemble Mr. Spock than Shakyamuni!   I think this is even further compounded by the Pali and Sanskrit languages which often use prefixed negations to express positive qualities (e.g. ahimsa or non-harming).  Based on that linguistic trope, it's very easy to think that Buddhism is only interested in a state in which negative qualities are absent.   

Not to get caught in the weeds over what Buddhism is after (and you might want to stay away from words like "eternal" on a Buddhism Q&amp;A site!), but let me assure your that liberation is anything but neutral.  There are positive states like the seven factors of enlightenment (of which "joy" is one!), the compassion, equanimity, etc. of the Brahmaviharas, the sweetness of meditative absorption, among many, many others that are of paramount importance and have to be cultivated with just as much zeal as one uses to overcome their negative views and behaviors.