"Hiking" the John Muir Trail in the winter is possible but it would be a major undertaking. First, most of the trail is usually under several feet (6' to 10') of snow by that time of the year (read the accounts of the Donner Party). So, route finding would be difficult at best and you would need skis or snowshoes to travel. Second, none of the resupply points are open in the winter so you would have to carry all of your food and fuel with you from the start. At 2 pounds of food per day, 27 days of food would weigh 54 pounds. You would also need a lot of fuel because you would often be melting snow for water. Third, it is cold in the Sierra in the winter. Check the weather statistics for Mammoth Lakes to get an idea of temperatures and snowfall and then remember that most of the JMT is as high or higher than is Mammoth. Fourth, you would need winter
mountaineering experience and equipment. Crossing Muir, Mather, Pinchot and Glen passes are well over 12,000 feet high while Forester Pass and Trail Crest Pass are both over 13,000 feet and have steep pitches. Finishing the trail on top of Mt. Whitney would take you to nearly 14,500'. Winds can be fierce at these altitudes. Winter in the High Sierra is not for sissies.
To camp and move you would need specialized knowledge and equipment. You should be comfortable with techniques such as knowing how to use crampons, skis and/or snowshoes and how to self-arrest from a fall. You would need a four season mountaineering tent, mattress and sleeping bag and the equivalent in clothing.
I seriously doubt that one could cover the full JMT in 27 days in the winter; it would require averaging over eight miles a day. This would be tough sledding
in the winter. . Were I you, I would wait and do the JMT in the summer.