I can't identify the components exactly, but I am fairly sure that they are simply diodes.
I think that there are two feeds of +12 V to the module, and each feed is connected to the side of one of those diodes that is away from the inlet connector, and connected to the anodes of the diodes. The cathodes are connected together, and that voltage is used to run the control part of the module, but not the power parts.
That allows the control part to run if either power feed is working. My guess is that the power after the diodes goes to the switch mode regulators that are the other side of the big toroidal inductors.
You can check the continuity between the cathodes and the power feeds of the module. If they don't connect, I've got it wrong.
You can probably use just about any rectifier diode in that package.
However, that doesn't explain why the diodes fail. It's clear that both are damaged. One is worse than the other but that could simply be a slightly different voltage on the two feeds to the module. Either diode would be rated to take the full load current for the control part of the module.
When the diodes have been replaced, if something else has gone wrong in the module, it will take an excessive current when powered up. It could be that there was something connected incorrectly outside the module that ended up taking too much current, in which case the module might be fine, except for the diodes. A fault on the parts after the switch mode power supplied wouldn't caused the diodes to burn.
I suggest you check the continuity as I may have the wrong idea about what the diodes do. If they are in the power supply to the control parts, change them, and power the module from a current limited supply, limited to half an amp or so. Obviously you won't be able to run the power parts under those conditions, but you should be able to see if the circuit functions.