On a unit cube, point N is an arbitrary distance n up the 1st diagonal, and M is a distance m up the second diagonal. Call the distance^2 between the points D2.  We have the function D2(n,m). Find m(n) that minimizes D2 for any n. Then find the minimum of this function, D2[n,m(n)], over n.    

 Using coordinates (x,y,z)

N = (n/sqrt2, n/sqrt2, 0), and

 

M = (1, m/sqrt2, m/sqrt2)

D2(n,m) = del_x^2 + del_y^2 + del_z^2 

             = 1 + n^2 - m n - sqrt2 n + m^2                 (eqn 1)

(d/dm) D2(n,m) = 2 m - n = 0 at the minimum 

m(n) = n/2

Plugging m=n/2 into eqn 1 gives the minimum separation for any n:  

D2(n) = 1 + n^2 - (n^2)/2 - sqrt2 n + (n^2)/4 

D2(n) = 1 + 3/4 n^2 - sqrt2 n                            (eqn 2)

(d/dn) D2(n) = (3/2) n - sqrt2 = 0 at the minimum

n = (2/3) sqrt2

Plugging this into eqn 2 give the minimum distance: 

D2 = (1/3), Dmin = sqrt(1/3)

And, yes, there is symmetry: one point is 1/3 the way up the diagonal, m = (1/3) sqrt2, and the other is 1/3 the way down, n = (2/3) sqrt2