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Let, the three consecutive positive integers be $x-1, x $ and $x+1.$

Now, $(x-1)(x )(x+1)=15600$

$\Rightarrow x(x^{2}-1^{2}) =15600$

$\Rightarrow x^{3}-x =15600$

$\Rightarrow x^{3}-x =(25)^{3}-25$

$\Rightarrow \boxed{x =25}$

So, we can get.

- $x-1=25-1=24$
- $ x+1=25+1=26$

$\therefore$ The sum of the squares of these integers $=24^{2}+25^{2}+26^{2}$

$\qquad \qquad = 576+625+676 = 1877.$

Correct Answer $:\text{D}$