It is worthwhile to note that $\Delta t' = \gamma \Delta t$ only for events with the same $x$ coordinate. The more general relation (i.e. Lorentz transformation) is:
$$\Delta t' = \gamma \left(\Delta t - \frac{v}{c^2}\Delta x \right)$$
where we assume that $v$ is the $x$-component of the velocity of the observer moving with respect to the mirrors, so that to that observer, the mirror setup is moving at $-v$.
By considering the time between successive $E_1$s (say), we can avoid having to consider $\Delta x = 0$, and in that case, we get, for the overall $E_1$ to $E_1$ cycle,
$$(2\Delta t') = \gamma (2\Delta t)$$
Otherwise, the time between $E_1$ and $E_2$ will be different for the 'onward' and 'return' journey. For light going from $M_1$ to $M_2$, assuming that $M_2$ is in the positve $x$-direction from $M_1$, $\Delta x = c\Delta t$, and since $\gamma = (1-\frac{v^2}{c^2})^{-\frac{1}{2}}$,
$$\Delta t'_{12} = \sqrt{\frac{c-v}{c+v}}\Delta t$$
For light going from $M_2$ to $M_1$, $\Delta x = -c\Delta t$, and
$$\Delta t'_{21} = \sqrt{\frac{c+v}{c-v}}\Delta t$$
The former goes to zero, and the later goes to infinity as $|v| \rightarrow c$, with $v \gt 0$.
For $v < 0$, this behavior is inverted.
Also note that in the moving observer's frame, the separation between $M_1$ and $M_2$ goes to zero. The situation looks like a pair of flat mirrors in contact with the light pulse at the interface, all travelling at the same speed. It takes no time for light to go from one mirror to another in one direction, but infinite time in the other. We can therefore get the same results by considering length contraction instead of time dilation (this may also be somewhat more intuitive here).
Therefore, if $M_2$ is in the positive $x$ direction from $M_1$, and $v \gt 0$, then it turns out that $E_2$ happens instantaneously after $E_1$, but it takes an infinite time for $E_1$ to occur again. This obviously changes (i.e. $E_1$ and $E_2$ may be exchanged) with the 'signs' of the mirror separation and observer velocity. This is one of the reasons that, for simple thought experiments with 'light clocks', the mirror separation is chosen in a direction perpendicular to the relative motion: so that arguments about 'time elapsed' are uniform.
As an additional remark, note that there is no problem with the overall idea that some things may be infinite when you approach the speed of light; that is built into special relativity almost by design.