If $x$ is simple, then this is true. Here is an outline of the proof. 

1) As $x$ and $y$ are freely homotopic, $y=x^g$ for some $g$. As $x$ is simple, $A_x$ and $A_y$ are disjoint.

2) Consider the Theorem 7.38.3  of Beardon's "Geometry of discrete group." Observe that $g$ is the common perpendicular, therefore $\epsilon =+1$.

3) This implies if length of $x$ increases, so does length of $x*y$ (in Teichnuller space).

4) Take right twists along the curve $z$. This will increase the length of $x$ (as $i(x,z)\neq 0$) and hence must increase the length of $x*y$ which implies $i(x*y,z)\neq=0$.

As long as $A_x$ and $A_y$ are disjoint, this proof will work. The other case is when $A_x\cap A_y\neq \emptyset.$ Then $x$ is not simple and their product formula is given by Theorem 7.38.3  of Beardon's "Geometry of discrete group," where $v_2$ will be a lift of the self-intersection point. Now there are two cases:


1) If $cos\theta$ is positive at $v_2$ then the above arguments hold true. 

2) If $cos\theta$ is negative, it is the same question as [this][1]mentioned by Ian Agol for the following reason. Take the self intersection point and consider the two branches of the curves starting and ending at this intersection point. Name them $x$ and $y$. If $y$ is a power of $x$ then you are done by your observation. If not then you get the case $n=1$ of the above question. 


  [1]: http://mathoverflow.net/questions/258189/geometric-intersection-number-for-product-of-elements-of-the-fundamental-group?noredirect=1&lq=1