Scale the square to $[0,1]^2$ and write $X=(x,y).$ The rays must include those through the four vertices. Every small triangle has area $1/n.$ The triangles whose bases partition the bottom side together have area $y/2,$ so their number is $ny/2.$ Similarly, the numbers along the top, left, and right sides are $n(1-y)/2,$ $nx/2,$ and $n(1-x)/2.$

These four numbers must be positive integers. Hence $n$ is even and $$\begin{gathered} X=\left(\dfrac{2i}{n},\dfrac{2j}{n}\right), \\ 1\le i,j\le\dfrac n2-1. \end{gathered}$$ Conversely, partitioning each side into the indicated number of equal segments and joining the division points to $X$ produces $n$ equal-area triangles. Thus these are exactly the partitional points.

For $n=100,$ the points are $(i/50,j/50)$ with $1\le i,j\le49,$ giving $49^2=2401.$ Such a point is also $60$-ray partitional exactly when $i/50=c/30$ and $j/50=d/30$ for integers $c,d.$ Thus $i$ and $j$ must both be multiples of $5.$ There are $9$ choices for each, so the overlap has $9^2=81$ points.

So the count is $2401 - 81 = 2320.$

Thus, the correct answer is C.