Euler characteristic of the simplicial complex of sets of elements in a semilattice with non-zero meet

To get a complete answer, I think you will need to use the specific properties of your semilattices. But I think the key observation is to notice what the maximal faces of your complex $Z_c$ are:

A subset $S$ of $L$ has meet $\bigwedge S>0$ if and only if there is some atom $a$ such that every element of $S$ is above $a$. (Here, "atom" means "element that covers 0", and the "meet" $\bigwedge$ is what you are calling "min".) Thus for every atom $a$, there is a maximal face of $Z_c$, consisting of $a$ and all the elements above it.

Since you know what the maximal faces are, you will probably want to see if you can use the combinatorics of your specific semilattices to make a ("nonpure") shelling of the complex in the sense of Björner and Wachs. If you find a shelling, you can compute the Euler characteristic of the complex using techniques in their paper "Shellable Nonpure Complexes and Posets. I." (Specifically, look at Section 3.)

This is a special case of the crosscut theorem. See e.g. Corollary 3.9.4 of Enumerative Combinatorics, vol. 1, second ed. Let $L'$ be $L$ with a top element $\hat{1}$ adjoined. In Corollary 3.9.4 take $X$ to be all elements of $L'$ not equal to $1$. We get $\chi=-\mu_{L'}(0,1)$, where $\mu_{L'}$ is the Möbius function of $L'$. There is lots of information about computing Möbius functions in the above reference.