Hi Beth.
They are pretty! Why on earth did the vet think they should be euthanized?? I think I'd be shopping for a new vet. Were you able to get any background info. on them besides estimated age?
Jack looks like a Cinnamon Pied White Face, and Diane looks like a Grey Pearl-Pied White Face. Do I have it correct? Sometimes in pictures it is hard to tell. Either way they should have some pretty babies as long as they aren't related.
Some tiels don't start sitting their eggs tight until they are done laying, some start setting right away. They could be inexperienced. Often the first clutch is unsuccessfull, so don't be surprised.
As far as the nest, I'd try to get a more closed in box filled at least 1/3 with all natural shredded aspen (prefered) or clean pine bedding. My birds seem to do well with that. Do NOT get cedar or the pine bedding with the green chlorophyll treated chips as they can be toxic to birds. Shredded paper is usually too light and the eggs sink to the bottom and become covered. I have never tried the paper hamster bedding so I can't comment on that.
If you have any other questions, feel free to ask.