Why is EXACT-CLIQUE not in co-NP?
The problem "Does the largest clique have a size ≤ k" is in co-NP - if the answer is NO then there is a clique of size > k, and guessing that clique proves that the answer is NO. 

But the problem here is "Does the largest clique have a size of exactly k". If the answer is NO then it could be that the largest clique has size k-1 and there are no cliques of size k, k+1, k+2 etc. Sure, you could guess a clique of size k-1 - but that doesn't prove a clique of size k doesn't exist. And you can't guess a clique of size k+1 because no such clique exists.