Linkage and Cohen-Macaulay-ness

The question is local. So, let $R$ be a local ring which is Gorenstein. $I,J\subset R$ define $Y,Z$ as in your question. Then you have an exact sequence $0\to I\to R\to R/I\to 0$ and we are assuming that $R/I$ is Cohen-Macaulay. Notice that all $R,R/I,R/J$ have the same dimension $d$. Dualizing, one gets $0\to\omega_{R/I}\to R\to R/J\to 0$. This implies that the depth of $R/J\geq d-1$. By going modulo a general set of $d-1$ elements in the maximal ideal, one can reduce to the case where $d=1$. Now dualize again to get, $0\to \operatorname{Hom}_R(R/J,R)\to R\to R/I\to\operatorname{Ext}^1_R(R/J,R)\to 0$. It is clear by naturality, that the map $R\to R/I$ is onto and thus the ext is zero. This says that depth of $R/J>0$ which is what we wanted.