Being "affine" in this case  does not make much sense,
because the hyperkaehler deformation is a complex manifold, without 
a fixed algebraic structure. Simpson produced an example of a 
hyperkaehler deformation of a space of flat bundles 
admitting several algebraic structures, both
inducing the same Stein complex structure; one of them 
is affine, another has no global algebraic functions.
In fact, the space F of flat line bundles on elliptic curve 
(with an appropriate algebraic structure, defined by 
Simpson) is an example of such a  manifold,
it is biholomorphic to $C^*\times C^*$, but 
this biholomorphic equivalence is not algebraic,
and F has no global algebraic functions.

However, you can show that a hyperkaehler deformation
of a resolution of something affine has  no non-trivial complex
subvarieties (arXiv:math/0312520), except, possibly, some 
hyperkaehler  subvarieties The latter don't exist, because 
the holomorphic symplectic form $\Omega$ on such a manifold
is is lifted from the base, which is affine, hence $\Omega$
vanishes on all complex subvarieties.

Therefore, a typical fiber of such a deformation is Stein. 

Indeed, a hyperkaehler deformation  of a 
resolution of something affine remains holomorphically convex. To see this
if you produce a function which is strictly plurisubharmonic outside of
a compact set (we have such a function, because we started from something 
affine), and apply the Remmert reduction.