1. Find the radius of the incircle of the equilateral triangle

For an equilateral triangle of side $a=12$, the inradius is

$r=\frac{a\sqrt{3}}{6}$

So,

$r=\frac{12\sqrt{3}}{6}=2\sqrt{3}$

2. Square inscribed in the circle

A square inscribed in a circle has its diagonal equal to the diameter of the circle.

Diameter of the circle:

$d=2r=4\sqrt{3}$

Let the side of the square be $s$. Then

$s\sqrt{2}=d=4\sqrt{3}$

Hence,

$s=\frac{4\sqrt{3}}{\sqrt{2}}=2\sqrt{6}$

3. Find area and perimeter of the square

Area:

$m=s^2=(2\sqrt{6})^2=24$

Perimeter:

$n=4s=4(2\sqrt{6})=8\sqrt{6}$

4. Compute $m+n^2$

$n^2=(8\sqrt{6})^2=64\cdot 6=384$

Therefore,

$m+n^2=24+384=408$

5. Compare with options

Thus the correct option is:

$\boxed{408}$

which is Option A.

6. Verification with stored answer

Stored correct answer: A

Our derived answer is also A.