How to design a simple current based class A amplifier
Since Hfe varies greatly with temperature and operating current your class A design is going to have quite variable gain.

The 2N3904 has high variance of Hfe with operating current, and requires substantial current to get good gain. 

Since you seem to have established the Zener current to produce the best noise output, it may be a circuit like this would be better. 







simulate this circuit &ndash; Schematic created using CircuitLab

R1/R2 provide about 1mA current and set the base close to 2V.
R4 has about 1.2V across it and set the operating point of the 2N3904 at about 18V.
R3/R4 set the gain to 10 for the noise signal.