smc

Let the three consecutive positive integers be $a-1$, $a$, and $a+1$. Since the mean is $a$, the sum of the integers is $3a$. So $8$ times the sum is just $24a$. With this, we now know that $a(a-1)(a+1)=24a\Rightarrow (a-1)(a+1)=24$. $24=4\times 6$, so $a=5$. Hence, the sum of the squares is $4^2+5^2+6^2=\boxed{77}$.