If you have a bantam hen laying, her egg will not get any bigger.
I guess it depends on you and the amount of space you have in your hutch, but I would say yes. If she is a bantam then a shoe box will work. Not saying that she will lay in it though.
Now as far as the time of day, well that just depends on the chicken. There is no set time limit that they "have" to lay. I have some that lay in the morning and some that lay in the evening. So you will just have to check from time to time and see if she has laid.
I hope this helps alittle.....