Easier than a pentakill

Numbers with 0,1,5 or 6 in units place will have squares that end in 0,1,5 or 6 respectively.

Any two digit number ending in 0 will have it's square ending in $\boxed{00}$ .

So, we don't check the squares of the numbers ending in 0.

Regarding numbers ending in 1, they will be of the form $10k+1$ where $k \in \{1,2,\ldots ,9\}$ and their squares will be of the form:

$\displaystyle 100 k^2 +10(2k) + 1$

Thus the unit's place does have 1 but the ten's place will have 2k or the unit's place of 2k. For example, $(10(2)+1)^2=21^2 =441$

So it does end in 1 but the ten's place has 4=2(2). One more example,

$(10(6)+1)^2= 61^2=3721$

Here the ten's place in 3721 is occupied by unit's place of 12(=2(6)).

So, we ignore these cases.

Now we just have to check the squares of 15,25... 95 and 16, 26.... 96 to get

$25^2 =6\boxed{25} \text{ \& } 76^2=57\boxed{76}$ .