For a ring $R$, the space $\operatorname{MaxSpec} R$ endowed with the Zariski topology has two evident properties: its singleton sets are closed (i.e., the T1 separation property) and it is quasi-compact. In his thesis, M. Hochster showed that conversely any quasi-compact T1 topological space $X$ is homeomorphic to $\operatorname{MaxSpec} R$ for some commutative ring $R$.
A totally disconnected topological space is automatically T1: closures of points are connected sets! So your question reduces to one of general topology: is there a totally disconnected, quasi-compact non-Hausdorff topological space $X$? The answer is yes and indeed Eric Wofsey's answer to your earlier math.stackexchange.com question provides an example.
In summary: no, not necessarily.