You may be thinking that from the description of this as a "flyback diode", that it is subject to a high voltage, such as in a CRT video monitor EHT supply.
As it happens, the very opposite is true - it is subject to no more than the supply voltage of the system you are using as it prevents excessive "flyback" when the relay is switched off. Nor does it have to be a high-speed diode because "speed" in diode terms almost exclusively refers to how fast they switch off and the very function here defines that it switches off very slowly!
Finally, its voltage drop is of little concern as you are not concerned about a volt or so in terms of the switching device's maximum rating (otherwise your design would be wrong), so a Schottky diode is of no benefit.
The only real requirement is that the current rating match that drawn by the relay. If that is less than 300 mA, a 1N914/ 1N4148 (technically rated at 4A surge and 100V) will suffice.