Square of the Pauli matrices and the identity matrix The square of any of the three Pauli Spin matrices is equal to the identity.
Is there any physical meaning to this?
Would you expect it?
Maybe in the context of the $SU(2)$ group?
 A: This is because there are just two possible values to the spin in any direction, $-\frac{\hbar}{2}$ and $\frac{\hbar}{2}$, they just differ in a sign, so when you square it you get a single value $\frac{\hbar^2}{4}$. Think about this, the only possible value when you measure the square of $S_z$ is $\frac{\hbar^2}{4}$ for any state, so
$$
<\psi|S_z^2|\psi>=\frac{\hbar^2}{4} \quad \forall \, |\psi>
$$
So it must be a multiple of the identity operator
$$
S_z^2=\frac{\hbar^2}{4} I
$$
Remember that $S_z$ is proportional to the Pauli matrices, $S_z =\frac{\hbar}{2} \sigma_z$.
A: OP asks:

Is there any physical meaning to this?

Yes, the Pauli matrix $\sigma_j$ represents (up to a proportionality factor) the spin in the $j$th direction of a spin $\frac{1}{2}$ system. Such system has only two spin states: $\uparrow$ and $\downarrow$, with opposite eigenvalues. The square $\sigma_j^2$ can no longer see the sign, so it only has one eigenvalue, cf. comment by BMS. In other words, the square $\sigma_j^2$ is proportional to the identity matrix.    
