First we'll need a few tensors.
A 6th-order tensor ${\mathbb M}$, whose components ${\mathbb M}_{ijklmn}$ are unity if $\,(i=k=m)$ and $(j=l=n),\,$ but zero otherwise.
This tensor makes it possible to rewrite a Hadamard ($\circ$) product using Frobenius (:) products like so
$$\eqalign{
A\circ Z &= A:{\mathbb M}:Z \cr\cr
}$$
Next we'll need two 4th-order isotropic tensors whose components are
$$\eqalign{
{\mathbb E}_{ijkl} &= \delta_{ik}\,\delta_{jl} \cr
{\mathbb B}_{ijkl} &= \delta_{il}\,\delta_{jk} \cr
}$$
These tensors make it possible to re-arrange matrix products
$$\eqalign{
A\,dX\,Z &= A{\mathbb E}Z^T:dX \cr
A\,dX^T\,Z &= A{\mathbb E}Z^T:{\mathbb B}:dX \cr\cr
}$$
Now we are ready to find the differential and gradient of your function
$$\eqalign{
F &= A\circ X^TX \cr &= A:{\mathbb M}:X^TX \cr\cr
dF &= A:{\mathbb M}:(dX^TX+X^TdX) \cr
&= A:{\mathbb M}:({\mathbb E}X^T:{\mathbb B}:dX+X^T{\mathbb E}:dX) \cr
&= A:{\mathbb M}:{\mathbb E}X^T:{\mathbb B}:dX + A:{\mathbb M}:X^T{\mathbb E}:dX \cr
&= A:{\mathbb M}:\big({\mathbb E}X^T:{\mathbb B}\,\,+\,\,X^T{\mathbb E}\big):dX \cr\cr
\frac{\partial F}{\partial X} &= A:{\mathbb M}:\Big({\mathbb E}X^T:{\mathbb B} \,+\, X^T{\mathbb E}\Big) \cr\cr\cr
}$$
Another approach is to use vectorization.
Let
$$\eqalign{
f &= \operatorname{vec}(F) \cr
x &= \operatorname{vec}(X) \cr
a &= \operatorname{vec}(A) \cr
{\mathcal A} &= \operatorname{Diag}(a) \cr
}$$
Then
$$\eqalign{
df &= a\circ\operatorname{vec}(dX^TX+X^TdX) \cr
&= {\mathcal A}\,\Big((X^T\otimes I)B\,dx + (I\otimes X^T)\,dx \Big) \cr\cr
\frac{\partial f}{\partial x} &= {\mathcal A}\,(X^T\otimes I)B\,+\,{\mathcal A}\,(I\otimes X^T) \cr
}$$where $B$ is the Kronecker Commutation matrix.
This is actually quite similar to the tensor result, with
$$\eqalign{
{\mathcal A} &\sim A:{\mathbb M} \cr
(X^T\otimes I)B &\sim I\,{\mathbb E}\,X^T:{\mathbb B} \cr
(I\otimes X^T) &\sim X^T{\mathbb E}\,I \cr
}$$
