To be close to second year undergrad physics, consider a non-relativistic electron with energy $E$, bounded to a double well potential $V({\bf r})$ with a classically forbidden tunneling region with potential energy $V_0$ in between, i.e., $E<V_0$. (Let us assume for simplicity that the full potential profile $V({\bf r})\leq V_0$, i.e., $V_0$ is a global maximum for the profile.)
(source: orst.edu)
Figure 1: An example of a double well.
As I read the question, Josh Chen is not disputing that an electron prepared in one well can reappear in the other well. Instead, the question is that since the integral of the square of the wave function over the classically forbidden tunneling region
$$ \int_{\{{\bf r}\in\mathbb{R}^3\mid V({\bf r}) > E\}} d^{3}r \ |\Psi({\bf r},t)|^2~>~0,$$
is strictly non-zero, does that actually mean experimentally that there is a non-negative probability to find the electron inside the classically forbidden tunneling region as the Born rule tells us, and how would one measure that probability, at least in principle?
Yes, the Born rule holds also in this situation. To measure the position of the electron, we will here use a photon with wave length $\lambda$ and of energy
$$E_{\lambda}~=~hf~=~\frac{hc}{\lambda}.$$
We will assume that the energies involved
$$|E|, |V|, E_{\lambda} ~\ll~ E_0=m_0 c^2$$
are much smaller than the rest energy $E_0$ of the electron, so we can treat the electron using non-relativistic quantum mechanics.
The electron's wave function $\Psi({\bf r},t)$ is exponentially decaying in the classically forbidden tunneling region with a characteristic tunneling penetration depth
$$\delta ~\sim~ \frac{h}{\sqrt{2m_0(V-E)}}~=~\frac{hc}{\sqrt{2E_0(V-E)}}.$$
(Since we are not really interested in the possibility that the electron could reach the other well, let us for simplicity assume that the electron penetration depth $\delta<\Delta$ is smaller than the separation $\Delta$ of the two wells, i.e. we are effectively studying a single well.) To use the photon as a 'microscope' in order to claim that we have detected the electron inside the classically forbidden tunneling region, the 'microscope' should have a resolution better than the electron penetration depth. In other words,
$$\lambda \ll \delta \qquad \Leftrightarrow \qquad E_{\lambda} \gg \sqrt{E_0(V_{0}-E)} $$
$$ \Rightarrow \qquad \frac{E_{\lambda}}{E_0} \gg \sqrt{\frac{V_{0}-E}{E_0}} > \frac{V_{0}-E}{E_0}
\qquad \Rightarrow \qquad E+ E_{\lambda}\gg V_{0},$$
i.e., the photon could knock the electron completely out of the well profile, so that the electron continues to spatial infinity. In principle, the incoming photon could be aimed at the classically forbidden tunneling region, and we could have prepared detectors in a $4\pi$ solid angle to capture and measure energy and momentum of all outgoing particles (the electron plus photons), and then calculate backwards to determine that a scattering event must have taken place inside the classically forbidden tunneling region. The missing energy between the incoming and the outgoing particles will be equal to the classically forbidden energy $E-V_{0}<0$.
On the other hand, if we had used soft photons with energy $E_{\lambda}<V_{0}-E$, the above inequalities get reversed, and the resolution will be too poor to determine whether the electron is inside or outside the classically forbidden tunneling region, cf. the uncertainty principle.
