Hausdorff Dimension of Arbitrary Julia Set

There is NO exact solution for the Hausdorff dimension of Julia sets for $z^2+c$ for general $c$. There are numerical ways to compute approximately (which I'm not vary familiar with), but the exact value is known only for very exceptional case.

Also, it is true that the Hausdorff dimension is equal to two for GENERIC c in the boundary of the Mandelbrot set: see Mitsuhiro Shishikura, "The Hausdorff Dimension of the Boundary of the Mandelbrot Set and Julia Sets".
Moreover, it is known that the area of the Julia set can be positive for some $c$: see Xavier Buff and Arnaud Chéritat, "Quadratic Julia sets with positive area" or by Artur Avila and Mikhail Lyubich, "Lebesgue measure of Feigenbaum Julia sets" (Note that the McMullen's paper on positive area is not for rational maps, but for sine family $f(z) = \lambda \sin z$ (and similar ones), which is a family of transcendental entire maps).

However, at the same time, there are many cases that the Hausdorff dimension is strictly smaller than two. For example, as already mentioned, if the critical points are non-recurrent, the Hausdorff dimension is strictly less than two: see Mariusz Urbański, "Rational functions with no recurrent critical points". And there are many $c$ in the boundary of the Mandelbrot set satisfying this property.