The string is more likely to break when the circle is vertical.
Work Step by Step
When the circle is horizontal, tension $T$ in the string is the only force contributing to the centripetal force $F_c$. The stone's weight $W$ is not related to $F_c$. When the circle is vertical, however, $T$ and $W$ interact to contribute to $F_c$. At the bottom of the circle, $T$ opposes $W$ so $F_c=T-W$ or $T=F_c+W$ As we can see, when the circle is vertical, there are points where $T$ has to both support $F_c$ and counter $W$, meaning its value is greater than tension $T$ when the circle is horizontal. The string is more likely to break as a result.