"Anyone can be enlightened" To my limited knowledge, in Zen, it is believed that anyone can be enlightened(if dedicated enough). 
Does everyone have this potential in other branches of Buddhism? 
Can everyone achieve this through meditation?
Is this what is meant by Buddha nature? 
Is it exclusive to humans(is language necessary?)
Buddhist philosophy, or even philosophy in general makes words difficult to use consistently. The concept of everyone is one of those words. Which makes the answer yes and no to the question: Can everyone be enlightened?

In, I believe, all schools and branches of Buddhism, all sentient beings can eventually gain enlightenment. With emphasis on "eventually". 


Animals are not able to, as long as they are an animal. 
Dwellers of lower realms, have an extremely hard time becoming enlightened so, and would probably need to accumulate some positive karma to become a human before they have a chance to. 
Beings of higher realms like deva's tend to have such a happy and fun life, practising is difficult, but not impossible. 
The human realm is seen as the best of all worlds in terms of practice.
Humans that have committed "Anantarika-karma" or Heinous Crime, (matricide, patricide, killing ahrat, wounding Buddha, creating a schism in the sangha) can no longer achieve enlightenment in this life.


Can everyone achieve this through meditation?

Now this is a difficult one. In the texts some of the enlightened students became so by listening to the words of the Buddha. Others became through practise. The main attributes needed seem to be a pure mind and insight, whether gained by meditation or otherwise.

Is this what is meant by Buddha nature?

I think it is, to an extend. This seems to be mainly a Mahayana concept and I'm not that familiar with it. I believe it's considered literally: "the potency to become a Buddha"
In the books I read of the Dalai Lama it is mainly revered to with respect to developing loving-kindness: "Training or developing your Buddha nature."

Is it exclusive to humans(is language necessary?)

Heavenly beings can also practise and become enlightened. I'm uncertain whether language is necessary, I can remember no sutta's about the subject.