If the pushbutton won't open the drives even when they're installed in another computer then it's clearly a problem with the drives. Perhaps a bit of a mystery as to why, but unfortunately facts are facts.
If that's the case, then I think you have three or four options:
1) Buy new drives. Expensive, but definitive.
2) Use the paper-clip technique to open the drive trays. Awkward, but free and it gets the job done.
3) See if you can find anyone to repair the drives. May not be any cheaper than new drives, and no guarantee of success.
4) Open the drives up yourself and tinker with them to see if there's a way to fix them. You might be lucky and see an obvious problem like a burned circuit trace that you can jumper with a wire. Or if you manually short the leads going to the button and the tray opens then you might be able to replace it the button with something else. Cheap, but labour intensive and no guarantee of success.
Something to keep in mind is that the lasers in burners have a limited life span. If these drives have been in service for a few years then they may be nearing the end of their life anyway. In that case investing more money in them may not be a good idea.