My take on egg eating and winter eggs:
"They say" that pullets will lay all winter without supplemental light.
I say, a lot depends on your lattitude, and the age of the pullets when they achieve sexual maturity in relation to the hours of available light, whether days are getting longer at that time or getting shorter at that time. Other factors to consider: breed of the bird, and strain of the bird within that breed. Protein content of the feed.
My best outcome re: having pullets lay during the winter without supplemental light was with pullets hatched in Feb. They achieved POL at around 16 weeks, before the summer solstice. Even at that, I only had one gal who continued laying all winter: an EE who gave me 3 eggs/week. I live at 44.5* lat. Folks south of that lat. have much better outcomes re: birds continuing to lay during the winter without supplemental light.
Since the second winter when egg production decreased to 0%, I have been providing supplemental light. I use my own protocol which is a combination of response to research and "what makes sense to me".
Warm spectrum light: research indicates that warm spectrum facilitates egg laying.
Allowing the flock to go through a natural decline in production which coincides with decreased day length and a molt. After my hens have entered or completed their molt, and egg production has decreased to as low as 0%, I start ramping up the light. I add about 1 - 2 hours/week, divided time between morning and evening. So, by the time they are at 14 hours/day, the light schedule looks like this. (on at 6:30 AM, off at 10 AM. On at 2:20 PM, off at 8:30 PM)
"They say": supplemental lighting should be provided in the morning so the birds go through a natural dusk, thus preventing the birds from going into panic mode when they are down on the floor doing chicken things when the lights suddenly go out at the end of the day. I say: this has never been a problem. No wild panic. Occasionally, there will be a slow learner. Generally, they learn to work with the lighting system and are up on the perches long before the lights go out.
In the past, I've used stick style solar landscaping lights, with a stick placed right below the supplemental light which is on a timer. This way when the big light goes out, the solar light provides some dim light for a bit. Not doing so this year, b/c I've not had a solar light which works well.
Re: egg eating. I say that egg eating is a response to end or beginning of laying cycle when shell glands are not functioning at their needed capacity. Increased problems could also be related to overall poor diet, lack of calcium, calcium intake out of synch with other needed vitamins and minerals. In particular, Vitamin D is necessary for calcium uptake. Other minerals also play a role in Ca++ uptake.
Egg eating occurs when shell quality is lacking. Other causes include exploratory pecking when young pullets are coming into POL, and are exploring the nest boxes and the contents of those boxes. Increase shell quality, and you will most often find that egg eating becomes a distant memory. (Oyster or egg shell, increased protein, especially animal protein, multi-vits)
Other causes of egg eating: crowding, birds who don't get opportunity to engage in normal chicken behavior. (give those birds extra greens, free range opportunities, sprouts, deep litter in coop and run, bales of hay to work on, multi level and other exploration opportunities in coop and run.) Or there may be one or more birds in the flock who have chronic metabolic deficiencies. This will present as a bird who consistently lays a weak shelled, leather or rubber egg, or an egg completely lacking membranes. Or it could present as a bird who simply does not meet her dietary needs as well as the other flock members. She is always seeking extra protein. I would cull either of these birds.