We've had something similar almost every late winter/early spring. Usually we leave them on the logs. Sometimes I cover with a tarp...blankets are good too. Covering helps if it is just a night or two before it warms again.
The main problem is that they quit growing unless it warms again fairly quickly.
This year we had a surprise flush in a warm wet spell and picked all that were large and opened. A few days later I picked the buttons because they were drying out and we expected more freezing temperatures.
Ours have never gotten mushy from freezing, they just quit growing and either dry out if there is no rain or mold if it rains a lot.
Rereading my post, I see I haven't given a definitive answer at all
It's just a judgement call every time, depending on the current weather and the forecast and the stage of the mushroom.