Unless she is broody she won't be interested in the eggs. Just take them and eat them, they are delicious. She will probably start laying an egg almost every day, mine does. And they go by the daylight hours not so much by temperature, the days are getting longer. All of my chickens and my duck are laying nearly every day now, but it's still fairly cold out. The box will be fine to move them in. You can use a crate or whatever is large enough for them. You may want to put a little bit of straw or shavings in the bottom.
