For any natural number $n=\sum_{i=0}^k a_i 10^i$ with $0\le a_i\le 9$, define $S(n)=\sum_{i=0}^k a_i$.
Then
$$
n^2=\left(\sum_{i=0}^k a_i 10^i\right)^2=\sum_{j=0}^{2k}\left(\sum_{i=0}^j a_i a_{j-i} \right)10^j,
$$
where $a_i=0$ if $i>k$.
Then
$$
S(n^2)\le \sum_{j=0}^{2k}\sum_{i=0}^j a_i a_{j-i} = \left(\sum_{i=0}^k a_i\right)^2=S(n)^2
$$
and we have equality if and only if $c_j:=\sum_{i=0}^j a_i a_{j-i}<10$ for all $j$.
It follows that if $S(n^2)=S(n)^2$, then $a_i<4$. In fact, if $a_j\ge 4$, then
$$
c_{2j}=a_j^2+\sum_{i=0}^{j-1}a_i a_{2j-i}\ge 16,
$$
which is impossible.
It also follows that if $S(n^2)=S(n)^2$, and some $a_j=3$, then $a_i\le 1$ for all $i\ne j$. In fact, if $a_j=3$ and $a_i\ge 2$ for some $i\ne j$, then
$$
c_{j+i}=2a_j a_i+\sum_{\underset{l\ne i,j} {l=0} }^{j+i}a_l a_{j+i-l}\ge 12,
$$
which is impossible.
If we now set $L(j)=\# \{a_i,\ a_i=j\}$ (which depends on $n$), then, if $S(n)^2=S(n^2)=100$, by the previous results necessarily we have
$$
(L(1),L(2),L(3))\in\{(10,0,0),(8,1,0),(6,2,0),(4,3,0),(2,4,0),(0,5,0),(7,0,1)\}.
$$
In principle you now can try these combinations in order to see which satisfies $c_j<10$ for all $j$.
For example, the smallest example with $(L(1),L(2),L(3))=(10,0,0)$ is $n=10\ 111\ 111\ 111$. Using an exhaustive search with Mathematica, one finds the smallest example with
$(L(1),L(2),L(3))=(8,1,0)$ is $n=1101111211$ (which is the example you mentioned and the absolute smallest example), and the smallest example with
$(L(1),L(2),L(3))=(4,3,0)$ is $n=1121102002$. Higher examples
require too much time using Mathematica, but I think that one can prove that the smallest example with $(L(1),L(2),L(3))=(0,5,0)$ is $n=2000020002022$. For this one
can use that if $S(n^2)=S(n)^2$, and for some $j_1<j_2<j_3$ we have $a_{j_1}=a_{j_2}=a_{j_3}=2$, then $j_2-j_1\ne j_3-j_2$. In fact, if $a_{j_1}=a_{j_2}=a_{j_3}=2$ and $j_2-j_1= j_3-j_2$
for some $j_1<j_2<j_3$, then
$$
c_{2j_2}=(a_{j_2})^2+2a_{j_1} a_{j_3}+\sum_{\overset {l=0}{l\ne j_1,j_2,j_3} }^{2j_2}a_l a_{2j_2-l}\ge 12,
$$
which is impossible.
I am also pretty sure that the smallest example with $(L(1),L(2),L(3))=(7,0,1)$ is $n=10101111013$ and that the smallest example with $(L(1),L(2),L(3))=(6,2,0)$ is $n=10101011122$. For $(L(1),L(2),L(3))=(2,4,0)$ I didn't found
(nor searched with purpose) a smallest example.
May be there are some other features than trying all possibilities by hand (or by computer).
${\bf{Edit:}}$ If you set $$L(k)=\min\{n: S(n)^2=S(n^2)=k^2\}$$
then
$L(1)=1$,
$L(2)=2$,
$L(3)=3$,
$L(4)=13$,
$L(5)=113$,
$L(6)=1113$,
$L(7)=11113$,
$L(8)=1011113$,
$L(9)=101011113$,
$L(10)=1101111211$,
$L(11)\le 1001101111211$.
