Why do Bell tests give perfect correlations? Suppose some decay process emits 2 electrons in opposite directions,
and their spin is measured by a Stern-Gerlach type device in a
particular direction, say Sz. The books say that if 2 detectors have
the same orientation, then the spin measurements are 100% correlated
(or anti-correlated). If one has Sz=+1/2, then the other has Sz=-1/2.
But what is the quantum state of the electrons before the measurements?
It seems to me that if it is anything but an eigenstate of Sz, then
the perfect correlations are impossible.
Is there any quantum mechanical explanation of the perfect correlations,
other than with an action-at-a-distance collapse of the wave function?
 A: Quantum mechanics has one peculiar and highly counterintuitive property:
If you know the complete state of a compound system $S$, this does not necessarily imply you know the complete state of either of the subsystems!
The quantum state you are after looks like this:
$$\frac{1}{\sqrt{2}} \left( \langle \uparrow \downarrow | - \langle \downarrow \uparrow| \right)$$
Since this is an entangled state, this means that you cannot decompose this into some product of the form 
$$ \langle \psi_A | \otimes \langle \psi_B | $$
In a way, it is therefore nonsensical to ask what the state of one of the electrons is! If you want to talk about one of the subsystems only, you must make use of the density-matrix formalism. An application of this would show that for the state I've depicted above, the individual spins are as undetermined as possible, being up or down with exactly equal probability.
Now to your second question. Sure, there are other explanations apart from the collapse of the wave function. But from a practical point of view, those are all equivalent and you can basically choose whichever you like best. There are the various flavors of the many-world interpretation, for example. 
EDIT In response to your comments:
Note that the correlation does not rely on the condition that the two measurements occur in such a way that light would be able to travel from one lab to the other in the time between the measurements! Indeed, if we call the two observers $A$ and $B$, it does not matter how far apart the labs are or when exactly the measurement takes place. How such experiments are carried out is that for both labs, a list of the spin is written down, which will read like $\uparrow \uparrow \downarrow \uparrow \downarrow \downarrow \dots$. After the experiment, $A$ and $B$ will compare their lists and find a $100\%$ correlation (well, barring some measurement errors that destroy coherence). 
I don't think it is possible to give a satisfying explanation relying on classical terms, because nature isn't classical.
I'm not sure I understand what you mean with "there's no eigenstate of $S_z$". The total state as I have written down is a linear combination of product states, which are formed from the basis states of systems $A$ and $B$. So of course, when $A$ or $B$ perform their measurements of the electron spin, they will find only eigenvalues of $S_z$. The important part is that the overall state of the system cannot be written as  a pure product state, no matter what basis you choose. This is the hallmark of entanglement, because if you had a product state $\langle \psi_A | \otimes \langle \psi_B$, a measurement on system $A$ would have no effect on the state of system $B$.
You can view the process of measuring the electron spin in systems $A$ and $B$ and then talking about the results as a measurement of the total spin's $z$-component. In terms of total angular momentum, the state I have written is equivalent to 
$\langle 0, 0|$, i.e. a state of total spin zero. Since this state is an eigenstate of the total spin with eigenvalue $0$, every measurement of the total spin's $z$-component must yield result $0$. 
A: But what is the quantum state of the electrons before the measurements? It seems to me that if it is anything but an eigenstate of Sz, then the perfect correlations are impossible.
Electrons have "wave particle duality" which sounds like philosophical garbage but is detectable in experiments such as this one. Rather than emitting the electrons in an unknown state, let's assume a device that specifically emits a pair of electrons that are 100% anti-correlated in Sx rather than Sz.
We will work in the usual Sz basis, and since Sx can be taken to be real in this basis, I'll ignore the difference between bra and ket vectors (and for convenience, write them all as bras). So one of the electrons is Sx+ = $(1,1)/\sqrt{2}$, the other is Sx- = $(1,-1)/\sqrt{2}$. The combined wave function is defined mathematically as a x product as in:
$$|x+-\rangle = (1,1)\times(1,-1)/2 = (1,1,-1,-1)/2.$$
But since electrons are indistinguishable, we could have the two electrons reversed. This would be the state:
$$|x-+\rangle = (1,-1)\times(1,1)/2 = (1,-1,1,-1)/2.$$
Note that the above two vectors are orthonormal.
Since electrons are fermions, we take the anti-symmetric combination. This means we take the difference between the above two joint wave functions:
$$|x+/-\rangle = (|x+-\rangle - |x-+\rangle)/\sqrt{2} = (0,1,-1,0)/\sqrt{2}.$$  

Now let's redo the calculation with two electrons that have spin Sz anti-correlated so that Sz+ =$(1,0)$ and Sz- = $(0,1)$. The two possible cases are:
$$|z+-\rangle = (1,0)\times (0,1) = (0,0,1,0)$$
$$|z-+\rangle = (0,1)\times (1,0) = (0,1,0,0)$$
and the anti-symmetric combination is:
$$|z+/-\rangle = (|z+-\rangle - |z-+\rangle)/\sqrt{2} = (0,-1,1,0)/\sqrt{2}.$$
This is the same as we got for the x case $|x+/-\rangle$ (other than an overall factor of -1 which is just the usual arbitrary complex phase).
Thus we see that according to the rules for quantum mechanics, it doesn't matter which two electrons we begin with. They could be spin in the x direction or z direction, or any direction; you get the same joint wave function.

Now the above explanation may be unsatisfying in that it is mathematical. If you want a more physical explanation for what is going on, maybe the following will help.
You can think of the electrons as being excitations, that is, as waves. When you measure a wave, you alter the wave and change it. In the case of the anti-correlated electrons, the combined wave has just enough stuff to give you two electrons with opposite correlated spin. It doesn't matter which way the spins are arranged, there's just enough to give you a pair of them.
One must be careful when one talks about the passage of time in these experiments. The measurement typically takes place long after the electron wave function has been divided into two portions, say one going to one detector, the other to the other detector. It might be several nanoseconds before an apparatus is able to permanently record which electron had which spin. During THAT (i.e. measurement) time, the electron's wave function has to be changed into another form, one which is suitable for measurement. This cannot happen instantaneously and it does not happen AFTER the electron wave function is split. Until the measurement is made, the electron, along with the measuring apparatus, is all in a Schroedinger Cat sort of combined wave function. Either possible result is possible, but the electron wave function only has enough juice to do one of the two possibilities.
The same "explanation" applies to things like Wheeler's "delayed choice" experiment. There actually is no delay in the choice in that the combination of experiment and particle are in a combined wave state until the measurement becomes permanent.
A: The correlations you describe seem to be trivial and completely explained by a conservation law (conservation of total Sz in this case). Classical example of  100% correlation: if a rocket with knowen total mass and velocity splits into to two parts then measuring the momentum of one of the parts gives the perfect knowledge of the momentum of the second part.
The fact that the particle you mention  are in a pure entangled quantum mechanical  state is not relevant to this 100% correlation. Correlations entering Bell inequalities are more subtle and necessarily involve measuring non-commuting observables.
