Even if your martagon seedlings are in pots outside, as long as they have been outside all fall (experiencing the downward trend of temps), they are ready to go through the cold winter period, regardless of what they look like, green or not. Throughout the plant kingdom (not just Lilium), there is often a phenomenon where seedlings (even up to around 5 years old in some cases) tend to be evergreen while mature plants of the same species are not. This is usually due to genetic juvenility factor(s) still being turned "on" in the young plant. Sometimes it is because of environmental reasons.
If your pots have been inside the whole time, then you'll need either refrigeratorlike temperatures for the duration of the winter, or begin with 2-3 weeks in the fridgelike temps before allowing to freeze in an insulated box in an unheated garage.
If your pots had been outside, but you brought them in when freezing weather was eminent, then they have already begun the transition to winter period. Put them in the fridge or in an insulated box in an unheated garage for the duration of the winter.
If the ground (or pot) will be frozen, than moisture content, as long as not overly wet, is inconsequential. But drier soil will help the green seedlings prepare for winter faster, even at 4°C. Optimally, it is best to stop watering and let them noticeably dry out, then enclose in a plastic bag so you won't have to worry about them getting too dry for the rest of the winter. Not that too dry soil will kill a lily (or even a seedling) in winter, it's just not optimal. I do this plastic bag scenario if I am holding anything in the fridge where water evaporates more quickly than if frozen.
Your seedlings that have "died off" are likely perfectly fine, and have gone dormant. It's another common phenomenon among geophytes that that the seedlings have a very abbreviated life above ground in their first season.
Edited for clarity.