Disproving or proving claim that if VCdim is "n" then it is possible that a set of smaller size is not shattered

I agree, the claim as written is incorrect. If $C^*$ is shattered by $mathcal{H}$, and $Csubseteq C^*$, then $C$ is also shattered by $mathcal{H}$; to be possibly over-pedantic:

For each $Bsubseteq C$, because then also $Bsubseteq C^*$, we have that there is an $Hinmathcal{H}$ such that $C^*cap H=B$. Note that this implies $Bsubseteq H$. Now, $Csubseteq C^*$ implies that $Ccap Hsubseteq C^*cap H$, so we have $Ccap Hsubseteq B$. And having both $Bsubseteq C$ and $Bsubseteq H$ implies that $Bsubseteq Ccap H$. So $B=Ccap H$.