Can we construct a binary tree with width and height Θ(n)?
Statements about $\Theta$-classes are statements about the behaviour of an algorithm/data structure, as (in this case) $n$ gets large. Thus, a single example can never prove such a statement. You'd have to provide a set that contains examples that become arbitrarily large and behave as desired. (The set, however, doesn't have to contain an example for every value of $n$.)

So you should try to construct such a set of examples. (And if you fail constantly, try to find out why. This might lead to a proof that no such set exists, i.e. the claim is false.)