smc

We know that the number of miles she drove is divisible by $5$, so $a$ and $c$ must either be the equal or differ by $5$. We can quickly conclude that the former is impossible, so $a$ and $c$ must be $5$ apart. Because we know that $c>a$ and $a+c\le 7$ and $a\ge 1$, we find that the only possible values for $a$ and $c$ are $1$ and $6$, respectively. Because $a+b+c\le 7$, $b=0$. Therefore, we have $$a^2+b^2+c^2=36+0+1=\boxed{37}$$