Since $ y = |x|, $ either $ x = y $ (if $ x \ge 0 $) or $ x = -y $ (if $ x < 0 $). In the first case, substituting into the first equation, we get $ y = y^2-6y+5, $ or \[0 = y^2-7y+5.\]This equation has the roots \[y = \frac{7 \pm \sqrt{7^2 - 4 \cdot 5}}{2} = \frac{7 \pm \sqrt{29}}{2},\]which are both positive and have a sum of $ 7 $. Since $ x=y $ in this case, the sum of the possible values of $ x $ is also $ 7 $. In the second case, substituting into the first equation, we get $ -y = y^2-6y+5, $ or \[0 = y^2 - 5y + 5.\]This equation has the roots \[y = \frac{5 \pm \sqrt{5^2 - 4 \cdot 5}}{2} = \frac{5 \pm \sqrt{5}}{2},\]which are both positive and have a sum of $ 5 $. Since $ x=-y $ in this case, the sum of the possible values of $ x $ is $ -5 $. We conclude that the sum of all possible values for $ x $ is $ 7 + (-5) = \boxed{2} $.