Z - X = (Y^2 - X)/(X*Y) is an integer, which then implies that

Z*X - X^2 = (Y^2 - X)/Y is an integer, which then implies that

Z*X - X^2 - Y = -X/Y is an integer.

Then this implies X is a multiple of Y; let X=K*Y.

Then back to the beginning: 

Z = X + Y/X – 1/Y 

Z = K*Y + Y/(K*Y) – 1/Y

Z = K*Y + 1/K - 1/Y

Now to satisfy the requirement for Z to be an integer, 1/K - 1/Y must be zero. (This is where the positive integer requirement speculated by broll comes into play, without that restriction then a nonzero 1/K-1/Y, like if Y=-2 and K=2, is possible.)

Then K=Y, which makes X=Y^2

Z = Y^2 + Y/(Y^2) - 1/Y

Z = Y^2 + 1/Y - 1/Y

Z = Y^2, a perfect square integer.

This verifies Jer's conjecture that X=Z=Y^2 forms the solution set (when the positive integer restriction is in place.)