Awfully sophisticated proof for the fact :) 
Just to relate it with the question about Knizhnik-Zamolodchikov equation:

http://mathoverflow.net/questions/95183/find-polynom-pz-with-values-in-cs-n-such-that-pz-sum-i-id1i-z-i

Consider the following KZ ODE:

$ p'(z) = \sum_{i=2...n} \frac{ Id + \pi( (1i) )}{z-z_i} p (z) $

As it is discussed in MO-question above it is known to have polynomial solution.

The reside at infinity is equal to  $Res=-\sum_{i=2...n} { Id + \pi( (1i) )}$.
Which is our beloved JM-element up to sign and n*Id.

Hence its eigenvalues must be non-positive integers (this is obvious since
at infinity the solution looks like $(1/z)^{Res}, so in order to be polynomial in z
they must be non-positive ints).
Hence we are done.

Moreover we got that eigs are greater or equal -n (as David Speyer proved directly above).