Roughly where are you located so we have a feel for how cold it gets and the length of your days and nights.
contemplating the pros and cons of adding supplemental light
Lighting is important to laying for different reasons. Light is one of the triggers to tell them when to release a yolk to start the internal egg-making process so the egg is laid during daylight hours so try to be consistent in what you do.
The actual length of daylight isn't that important, whether the days are getting longer or shorter is the really important part. The longer the day the more time they have to eat and digest food, but they tend to eat for caloric content so unless your days are ridiculously short they should get enough to eat.
Their normal cycle is to lay eggs and raise chicks in the spring and summer when the weather is nice and food is plentiful. When the days get short in the fall/winter they stop laying eggs and use the nutrition that was going into eggs to molt and replace worn out feathers. Then when the days started getting longer and weather got better they started hatching chicks again.
When we domesticated chickens we bred them to lay more eggs, maybe not go broody as often, and such but they still follow the basic pattern of molting when the days get shorter. The days getting longer can trigger them into laying but with the breeding to increase laying some don't need the days to get longer to start them laying. More than once I've had some pullets start laying early December during the shortest days of the year and days not getting longer. That was in Northwest Arkansas with day length just under 10 hours. I do not use supplemental light. I've had hens that finished the molt in late summer start laying in late October with the days getting shorter and not wait for the longer days of spring. There is a tendency on this forum for people to believe each and every chicken and the circumstances they are in to be identical across the globe. That's not close to the truth. No one can know when they will start laying or stop.
Since the principle criteria is whether the days are getting longer or shorter instead of day length you need the days to get longer to help kickstart laying. You want the days to not get shorter so you don't trigger a molt. But daylength is constantly changing. If you only provide extra light at one end of the day you may need to stay on top of that and adjust as you go. Some people adjust only in the morning, some only at night, and some do at both morning and night. Some people say if they turn the lights out suddenly the chickens don't get on the roosts in time. Others say they don't have that problem. I guess that shows that each situation is unique, we don't all get the same results.
Since days getting shorter can trigger a molt, if you turn your lights off before the natural daylight reaches that length of time you might trigger a molt. Say you establish and maintain 12 hours a day as your length of daylight, if you turn the supplemental lights off before the Equinox when the days are 12 hours long, you might trigger a molt. Not always but it can happen.
Since the criteria is days getting longer, it is often recommended to increase daylength gradually instead of one big jump. A lot of timers are set on 15 minute increments. A recommendation is add 15 minutes of light maybe twice a week until you get to where you want to be.
Do not expect instantaneous results. For a pullet or hen to go from not laying to laying mode she has to change her internal plumbing and grow some ova to the size of an egg yolk. That does not happen overnight, it can take weeks.
I agree with Aart. I would not start extending light for another month because of the age of yours. By then you are at the Winter Solstice and days are going to start getting longer, but that daily change at first is really slow. Extending the lights could help them get started earlier.
Hens can get worn out from egg laying constantly. If they go too long laying without a break egg quality and productivity can drop significantly. This is usually after more than a year. They need to stop laying and molt to recharge their body, then they are ready for another productive laying season. Lots of people extend lights and deal with that in their own way, we don't always get identical results. That's a big reason I often suggest you make decisions based on what you see not what some stranger over the internet like me tells you that you will see. We don't all see the same thing. One possible strategy is to freeze eggs in the good times to get you through the slim times.
and nest boxes to their coop next month
Personally I'd install the nests now. You never know when they will actually start laying, I want the nests available when they do. I don't want to train them to lay somewhere else because the nests aren't available when they need them.
But there is another reason. Some people on here will tell you that they might possibly, could, some of the time sleep in the nests if they are available. They are right, it is possible, it does sometimes happen. I want to know if they are going to sleep in the nests so I can fix the problem of why they are sleeping in the nests before I start getting poopy eggs. Sometimes the fix is easy, sometimes it takes work and time. Often they don't sleep in the nests at all. If yours do start sleeping in the nests come back on here and we can chat about it.
Secondary question . . . is there any way to prevent frozen eggs in the coop? Just collecting more frequently, I imagine?
What kind of temperatures are you talking about and for how long? What do your nests and coop look like? Photos could really help. It might help somebody come up with specific suggestions for your unique situation.
Collecting more frequently is an option if you are around. You may be able to safely heat the nests or at least some of them. Using an insulating nest bedding can help. Insulating around a nest may help, but you don't want then eating the insulation. You may find it isn't as big of a problem as you think but for some people it is.
But I do know you can't just put a light in the coop and all is good and well, I believe you have to adjust it at 15min per day intervals. As far as laying goes, they may, they may not, it's all up to the individual bird. I had 3 pullets a couple years ago that were hatched in August and started laying in January, so yes, it's possible. I just make sure I collect daily in the winter to avoid frozen eggs, although for the most part, normally we don't have a lot of freezing weather. 
