Is the mirror of a hyperkaehler manifold always a hyperkaehler manifold? Is the mirror of a hyperkaehler manifold always a hyperkaehler manifold? 
What I know so far is as follows:
In this paper (https://arxiv.org/pdf/hep-th/9512195.pdf) by Verbitsky, it is claimed that every hyperkaehler manifold is mirror to itself (which would meant the answer to my question is 'yes'). 
On the other hand, from the answer to this question - Mirror symmetry for hyperkahler manifold , it seems that this is not always the case, i.e., a K3 surface can have a nontrivial mirror. 
 A: I think Verbisky proves a refined form of the Mirror Conjecture for hyperkaehler manifolds, not the conjecture in the strict form. This is explained at page 3 of the paper that you link. 
In fact, the assumptions for the conjecture are known to hold only for a hyperkaehler manifold which is generic in its deformation class. More precisely, Verbitsky proves the following result, see page 26.

Theorem 5.4. Let $M$ be a compact holomorphically symplectic manifold, which is generic in its deformation class. Then the Mirror Conjecture
  holds for $M$, which is mirror dual to itself.

For instance, $\mathrm{Pic}(M)=0$ is enough, see the footnote in the same page. 
A: The following example from Hausel–Thaddeus say that mirror hyperKahler is HyperKahler. Lets explain it 
Let $\mathcal M$ be the moduli space of stable $GL_n$-Higgs bundles, non-singular
and hyperkahler and 
$\tilde {\mathcal M}$ moduli space of stable $SL_n$-Higgs bundles, non-singular and hyperkahler and ${\hat{\mathcal{M}}}:= \tilde{\mathcal M} /Γ $ is $PGL_n$-Higgs moduli space which is an orbifold($\Gamma\cong \mathbb Z_n^{2g}$)
Hausel–Thaddeus , proved the following theorem about Strominger-Yau-Zaslow conjecture for the hyperKahler mirror pair $(\hat{\mathcal M}, \tilde {\mathcal M})$
\begin{array}
^\tilde{\mathcal M} & \stackrel{}{\longrightarrow} & \hat{\mathcal M}\\
\downarrow{\tilde\chi} & & \downarrow{\hat\chi} \\
\mathcal A  & \stackrel{\cong}{\longrightarrow} & \mathcal A
\end{array}
The generic fibers $\tilde\chi^{-1}(a)$ and $\hat \chi^{-1}(a)$ are dual Abelian varieties. The pair of hyperkahler manifolds $(\tilde{\mathcal M} , J)$ and $(\hat{\mathcal M} , J)$ satisfy SYZ conjecture and mirror to each other
See 
Mirror symmetry, Langlands duality, and the Hitchin system, Inventiones mathematicae, July 2003, Volume 153, Issue 1, pp 197–229
Hausel - Thaddeus  interpreted SYZ conjecture in the context of Hitchin system and Langlands duality. Let briefly explain it
Let $\pi : E \to Σ$  a complex vector bundle of rank $r$ and degree $d$ equipped with a hermitian metric on Riemann surface $\Sigma$ . Take th moduli space 
$$\mathcal M(r, d) = \{(A, Φ) \text{ solving }(\star)\}/\mathcal G $$
(which is a finite-dimensional non-compact space carrying a natural hyper-Kähler  metric)
where 
$$F^0_A + [Φ ∧ Φ^∗] = 0 ,\; \;  \bar ∂AΦ = 0\; \; (\star)$$
Here $A$ is a unitary connection on $E$ and $Φ ∈ Ω^{1,0}(End E)$ is a Higgs field. $F^0$ denotes the trace-free part of the curvature and $\mathcal G$ is the unitary gauge group.
$\mathcal M(r, d)$ is the total space of an integrable system(which can be interpreted by the non-abelian Hodge theory due to Corlette), the Hitchin fibration, together with Langlands
duality between Lie groups provides a model for mirror symmetry in the Strominger-Yau and Zaslow conjecture. 
