I really really don't want to be doing lots of annoying algebraic manipulation, so I'll see if I can get there by reasoning alone.
None of x,y,z is equal to zero as xyz =1.
The fractions imply that none of x,y,z is ever equal to one because otherwise the fractions would be undefined. This means that at least one of x,y,z < 1 and at most two of x,y,z <1. Also, this means that at least one of x,y,z >1.
Let x < 1 (one of them has to be 0 I'm just arbitrarily choosing x). Then either x <= 0.5 or x > 0.5.
x<= 0.5 implies |x-1| >= 0.5 and so the fraction x^2/(x-1)^2 will have a maximum value of 1 and a minimum value that tends to but can never be zero.
Let z >1. Then z^2 > (z-1)^2 for all such z and tends to it's maximum value as z tends to one. However, z^2/(z-1)^2 is always greater than one and the larger z gets, the closer the fraction tends to one.
It doesn't really matter whether y is less than or greater than 1 because it makes no difference: the properties of z will remain unchanged. If the fractions in x and y are so small that they are practically zero, then the value of z will be so big that the fraction associated with it will tend to one (but will always be greater than it).
The inequality is therefore true.