Are reverse biased transistors stable?
No they are not stable.  


but stability is improved at lower currents &lt;=10uA from my research. *


The current gain Ic/Ib of bipolar transistors strongly decreases
when the oxide over the emitter-base junction is damaged. The results obtained with different stress conditions lead to point out two degradations processes: the main process consists in the increase of the recombination
in the vicinity of the space-charge region in the emitter-base junction.

Ref DEGRADATION OF JUNCTION PARAMETERS OF AN ELECTRICALLY STRESSED NPN BIPOLAR TRANSISTOR

N. TOUFIK, F. PILANCHON and P. MIALHE*

The above report was based on the 2N2222 which has a lower reverse breakdown threshold than the 2N3904 and also more noise voltage, but perhaps faster degradation. I have yet to find a report on the 2N3904 so an uncontrolled avalanche current is unstable with a small resistor 

Transistor I=1μA 10μA  100μA   Noise(I=10μA) For I=[5-50]μA noise:
BC107B  9.30V   9.30V   9.28V   200mVpp     inc., max=10μA, dec.
BC548A  8.44V   8.46V   8.45V   100mVpp     constant
BC547B  8.23V   8.22V   8.23V   100mVpp     decreases
BC547C  8.36V   8.35V   8.34V   120mVpp     decreases
BC546B  8.19V   8.21V   8.19V   120mVpp     inc., max=10μA, dec.
2N3904  10.82V  10.80V  10.76V  400mVpp     constant
2N2222A 7.20V   7.25V   7.23V   440mVpp     constant


Ref http://holdenc.altervista.org/avalanche/index.html


 The above uses a constant current source unlike Paul Campbell's rng2.0 which relies on avalanche current in base * hFE limited by Collector R = 3.3k to 20V so if hFE is only 100, diode can be (20V-1Vsat)/3.3k/100(hfe)= 58uA which would be excessive.



Given the initial question design saturates the noise into a digital signal and noise can vary in current with temperature as well as spectral bandwidth it is not stable and the threshold above 10uA where failure rapidly degrades is unknown, limiting the reverse current to the Mfg's test criteria for Vr would make most sense.

The next question is how stable does it need to be? What is life span criteria and repeatability to NIST scores? What are the test results over ambient range?

Increasing the coupling capacitor size may increase the max number of consecutive 1's, a figure of merit by extending the number of decades of frequency range at the expense of startup delay or sensitivity to supply regulation drift.