Any diode will do it, as long as all 4 of them are as identical as possible. Those are Ge diodes (and thus the +/-0.7V range == 2 x Vf for Ge) but Si should work just fine. The circuit is supposed to be symmetrical in order for the DC output to be linear with the ratio between the two vial's capacitive arms.
Another thing, the device output voltage is linear only around zero, or as long as the output DC is very small in comparison with Vf. For bigger capacitance differences between the vial arms, it won't be linear at all.
There will always be an AC component over the DC one, but the AC is supposed to be attenuated by the low pass RC filter on the right end of the schematic (0.1uF and 100K).
Another thing to consider is the impedance of a 0.01uF at 300KHz, which would be about 53 ohms. It is 2 orders of magnitude lower than the 5K6 resistor series with the bridge. If we set the generator for 300KHz and 5Vpp, this will be about 50mV of AC across the diode bridge. Small enough to consider any diode as a linear device for a swing of only 50mV around whatever decent DC static point. This small enough swing (and the symmetry of both the voltage waveform and the bridge arms) will make the whole thing a nice charge averaging circuit (one vial arm will have a bigger C because of the vial's tilt, thus its corresponding capacitor will accumulate a bigger charge then the other arm, because everything else is supposed to be symmetric). The two charges Qleft arm and Q right arm of the vial will end up averaged by the 0.01uF put across the bridge.
The first 0.01uF, in series with the generator is there just to be sure no DC possible offset from the generator will go into the bridge, and vice-versa to be sure no DC generated by the bridge will go back into the generator (a DC voltage pushed back into the generator's output would be bad, because it might affect the waveform symmetry for example - depends of the generator, but most of all we don't want any DC offset from the generator to go to our Vout). Symmetry in both the waveform and the bridge arms (identical transfer functions for all 4 diodes) are essential for this circuit. The waveform itself is not important as long as it's symmetrical and has no DC.
The 0.01uF inside the bridge is to integrate the charge imbalance over many periods of the 600KHz bridge switching. Ideally, the output voltage would be to be read across this diagonally 0.01uF inside the bridge, yet the author choose to read the DC voltage over the whole RC (5.6K and 0.01uF), thus adding the full 5Vpp swing from the generator over the DC we need. This full AC will be diminished by the 100K and 0.1uF RC low pass filter at the output.
My guess is this AC pollution was done to keep a common ground between the AC generator, the ground electrode of the vial and the DC voltmeter (for better shielding and noise rejection of the whole thing). If it were to have a floating DC voltmeter, will be much better to read the voltage across the bridge directly, without including the series 5k6 resistor.
Forgot to mention, the vial arms' capacitances are suppose to be very small in comparison with all other capacitors in the schematic, probably pF if not fractions of pF (no idea how small, didn't plugged any numbers). The absolute value is not important, only the ratio C_left/C_right of the vial.
Later edit:
Trying to proof reading the text afterwards, and realized that, without proper schematic and numbering of the components, it is all just a big TL;DR.
https://www.docdroid.net/enKb2ed/nasaames-simpletiltmeter.pdf