Having a particle entering the apparatus with spin state $|+\rangle$, for which $\hat S_x|+\rangle=+\frac\hbar 2|+\rangle$, I have a question about how to express the spin state when it comes out. I mean: will the spin state be an eigenstate of the spin $z$ component? or would it be a superposition of the $z$ component's eigenstate ?
A Stern-Gerlach apparatus oriented along the $z$ axis acts as a measurement on the basis of the $z$ component of the particle's spin. What that means is that the particle will always come out in an eigenstate of $\hat S_z$. You don't know which one, of course, as that is decided probabilistically.
Even if you don't actually observe the outcome of the measurement (in which case the system would be in the corresponding eigenstate), it is not accurate to say that the particle comes out in a superposition of $z$ eigenstates. While what comes out is a probabilistic mixture, it is a mixed state in which the coherence between the two contributions has been lost.
It's hard to believe that a correspondent as knowledgeable (and as highly rated) as Emilio Pisanty could show such a fundamental misunderstanding of quantum mechanics as he does in his answer to this question. If it were possible to send a single silver atom through an inhomogenous magnetic field of the type alluded to in this question (that is what we understand by the phrase "a Stern-Gerlach apparatus"), then if it enters the apparatus with a "sideways" spin, then it leaves with a net "sideways" spin (in the xy plane). There is no torque exerted by the magnets on the atom which is capable of flipping the spin into the z orientation, either z-up or z-down.
By the way, the notion that there is a machine which you can send a silver atom through and have it come out either spin-up or spin-down...well, no such machine exists. That's not what the Stern-Gerlach experiment did. I explain more about these things in this blogpost: The Quantization of Spin Revisited
DISCLAIMER: I am a recognized crackpot in this site whose answers are routinely and massively downvoted by people who know much more than me.