SuperSquares

First, make a list of all perfect squares of two digits $16,25,36,49,64,81$ , then we can analyse case by case:

If $AB=16 \Rightarrow BC=64 \Rightarrow CD=49\therefore 1649$ is a supersquare number

If $AB=25$ you cannot complete the number

If $AB=36 \Rightarrow BC=64 \Rightarrow CD=49\therefore 3649$ is a supersquare number

If $AB=49$ you cannot complete the number

If $AB=64$ you cannot complete the number

If $AB=81 \Rightarrow BC=16 \Rightarrow CD=64\therefore 8164$ is a supersquare number

The sum of all supersquare numbers is $\boxed{1649+3649+8164=13462}$

Nice problem!