clc, clearvars

intCt=0;

r=sqrt(sym(441000));

for x0=0:floor(r)

    x=sym(x0);

    y=sqrt(r*r-x*x);

    if y==round(y)

       disp([x round(y)]) 

    else

       intCt=intCt+floor(y);

    end

end

disp(intCt*4+1)

finds points

[210, 630]

[378, 546]

[546, 378]

[630, 210]

that the circle passes through

That's just the first quadrant, so there are 16 lattice points exactly on the circumference of the circle.

While it was at it, not asked for in the puzzle, the program counted 1,378,381 lattice points strictly within the circle. Well actually it counted  344,595 points on the positive y axis and strictly in the first quadrant not on any axis or circumference, which I multiplied by 4 and added 1 for the origin.