Could you say that two electrons in the ground state of a helium atom experience quantum entanglement? They are both in the same energy level and cannot have the same quantum numbers. If one is spin up, the other must be spin down. So, if one "flipped" spins, the other would have to also flip spins or violate sassy Pauli exclusion principle.
[Physics] Entanglement and the Pauli exclusion priciple
pauli-exclusion-principlequantum mechanicsquantum-entanglementquantum-spin
Related Solutions
The spin wave function is symmetric with respect to the exchange of particles. Therefore the spacial wave function has to be antisymmetric. I.e. at least one of the quantum numbers has to be different.
The wave function may look as if the electrons have opposite spin, but actually the spins are the same if measured at an axis 90° from z.
EDIT:
The spin eigenvectors of different axes are not independent from each other. $$ \begin{align} \left|\strut\uparrow\right\rangle &= \frac{1}{\sqrt 2}\left(\left|\strut x_+\right\rangle + \left|\strut x_-\right\rangle \right)\\ \left|\strut\downarrow\right\rangle &= \frac{1}{\sqrt 2}\left(\left|\strut x_+\right\rangle - \left|\strut x_-\right\rangle \right) \end{align} $$ Substitute that into your definition of $\left|\strut\psi\right\rangle$ and you will get $$ \begin{align} \left|\strut\psi\right\rangle &= \frac{1}{2\sqrt 2}\left[\left(\left|\strut x_+\right\rangle + \left|\strut x_-\right\rangle \right)\otimes\left(\left|\strut x_+\right\rangle - \left|\strut x_-\right\rangle \right) + \left(\left|\strut x_+\right\rangle - \left|\strut x_-\right\rangle \right)\otimes\left(\left|\strut x_+\right\rangle + \left|\strut x_-\right\rangle \right)\right]\\ &=\frac{1}{2\sqrt 2}\left[% \left|\strut x_+\right\rangle\left|\strut x_+\right\rangle% - \left|\strut x_+\right\rangle\left|\strut x_-\right\rangle% + \left|\strut x_-\right\rangle\left|\strut x_+\right\rangle% - \left|\strut x_-\right\rangle\left|\strut x_-\right\rangle% + \left|\strut x_+\right\rangle\left|\strut x_+\right\rangle% + \left|\strut x_+\right\rangle\left|\strut x_-\right\rangle% - \left|\strut x_-\right\rangle\left|\strut x_+\right\rangle% - \left|\strut x_-\right\rangle\left|\strut x_-\right\rangle% \right]\\ &=\frac{1}{2\sqrt 2}\left[% \left|\strut x_+\right\rangle\left|\strut x_+\right\rangle% - \left|\strut x_-\right\rangle\left|\strut x_-\right\rangle% + \left|\strut x_+\right\rangle\left|\strut x_+\right\rangle% - \left|\strut x_-\right\rangle\left|\strut x_-\right\rangle% \right]\\ &=\frac{1}{\sqrt 2}\left[% \left|\strut x_+\right\rangle\left|\strut x_+\right\rangle% - \left|\strut x_-\right\rangle\left|\strut x_-\right\rangle% \right]\\ \end{align} $$
To check for the symmetry you don't need this calculation. It is sufficient to check that $$ \frac{1}{\sqrt2}\left( \left|\strut\uparrow\right\rangle\left|\strut\downarrow\right\rangle + \left|\strut\downarrow\right\rangle\left|\strut\uparrow\right\rangle \right) = \frac{1}{\sqrt2}\left( \left|\strut\downarrow\right\rangle\left|\strut\uparrow\right\rangle + \left|\strut\uparrow\right\rangle\left|\strut\downarrow\right\rangle \right) $$
The Pauli principle says that wave functions have to be negated when swapping any two Fermions.
To a reasonable approximation the protons and neutrons in a nucleus occupy nuclear orbitals in the same way that electrons occupy atomic orbitals. This description of the nucleus is known as the shell model. The exclusion principle applies to all fermions, including protons and neutrons, so the protons and neutrons pair up two per orbital, just as electrons do. Note that the protons and neutrons have their own separate sets of orbitals.
I say to a reasonable approximation because neither nuclear orbitals nor atomic orbitals really exist. The atomic orbitals we all know and love, the $1s$, $2s$, etc, appear in an approximation known as the mean field. However the electron-electron pair repulsion mixes up the atomic orbitals so strictly speaking they don't exist as individual separate orbitals. This effect is small enough to be ignored (mostly) in atoms, but in nuclei the nucleons are so close that the nuclear orbitals are heavily mixed. That means we have to accept that the shell model may be a good qualitative description, but we have to be cautious about pushing it further than that.
Best Answer
In this context it is convenient to look at entanglement as a resource for quantum information tasks. There are several opinions about usefulness of correlations between identical particles as such a resource, but I think the most orthodox one is expressed in this review: http://arxiv.org/abs/1312.4311. The main point follows already from the title: you can extract entanglement from such correlations, but you can't use it otherwise, which means that there is no entanglement.