In this paper, we study the integral Cayley graphs over a non-abelian group $U_{6n}=\langle a,b\mid a^{2n}=b^3=1, a^{-1}ba=b^{-1}\rangle$ of order $6n$. We give a necessary and sufficient condition for the integrality of Cayley graphs over $U_{6n}$. We also study relationships between the integrality of Cayley graphs over $U_{6n}$ and the Boolean algebra of cyclic groups. As applications, we construct some infinite families of connected integral Cayley graphs over $U_{6n}$.