Nuno's answer to Any two points in a Stone space can be disconnected by clopen sets uses (and proves) the following:
Theorem: Let $X$ be a compact Hausdorff space. Then the quasicomponents of $X$ are its components. That is, its quasicomponents are connected.
Unfortunately, I'm having trouble grasping that proof, although each detail is fairly easy to follow. I'm wondering what other approaches there might be. I unwisely tried to find out by offering a bounty on that question; if anyone comes up with a good answer to this one, we can make an arrangement.
A couple thoughts I haven't been able to find a way to find a way to use, and which may not be viable:
Prove (somehow) that if $X$ has a disconnected component then it is not maximal compact.
Consider the sets of components and of quasicomponents as quotient spaces and work some sort of magic. They're both compact, and the set of quasicomponents is Hausdorff, but I don't see how to prove the set of components is better than $T_1$, which doesn't prove compact subsets closed.
Use characterizations of continuity (for connectedness) and compactness in terms of filters or nets.