The category $\mathcal{C}_X$ is complete and cocomplete if and only if $X$ is locally compact.
As you pointed out yourself, this category is always a preorder. I will first show that it is equivalent to the poset of closed equivalence relations on $\beta X\setminus X$ if $X$ is locally compact:
As soon as $X$ is Tychonoff, every compactification $Y$ of $X$ will be the image of a surjective map $f:\beta X\to Y$ and as such is fully determined by its restriction to the boundary $\beta X\setminus X$. If $X$ is even locally compact, it will be open in each compactification, so that each boundary $Y\setminus X$ is itself compact and thus given by a quotient of $\beta X\setminus X$. Conversely, every Hausdorff quotient of this Stone-Čech-Boundary then gives a compactification in this way.
Therefore, the category $\mathcal{C}_X$ is equivalent to the partial order of closed equivalence relations on $\beta X \setminus X$ with the natural ordering of set containment. This category is indeed complete and cocomplete: The infimum of a set of closed equivalence relations is just their intersection, while the supremum is given by the closure of the equivalence relation generated by their union.
It remains to see that the assumption of local compactness was really necessary. At first it seems promising that $X$ needs this property for the one-point compactification to feature (as the terminal object) in $\mathcal{C}_X$. But this lack is not yet a proof, since there might perhaps be some other, nonobvious, terminal object in this case. However, this is not the case: Observe that, for any two points $x,y\in \beta X\setminus X$, the quotient space $\beta X/x=y$ is still in $\mathcal{C}_X$. (This is straightforward to check.) But then, the supremum $Y$ of all these compactifications, were it to exist, would contain only one point beside $X$ (since all others must be identified). But this cannot happen unless $X$ was locally compact to begin with, so this supremum does not exist.
