The winter solstice and shortest day of the year in Boston is on Sunday, December 21, 2014 at 6:03 PM. The sun will rise at 7:10 AM and will set at 4:15 PM, making the day 9:04:34 long.
However, the 21st does not contain the earliest sunset and latest sunrise. The earliest sunset already occurred on December 8th and 9th when the sun set at 4:11 PM. The latest sunrise will actually occur from January 2nd-5th when the sun will rise at 7:14 AM.
Timedate.com has a sun calculator where you can see how many hours of daylight there is in Boston at any given day.
If you are looking for some winter solstice activities, you can go to the Isabella Stewart Gardner Museum on Thursday for their winter solstice party or check out the Revels winter solstice celebration until December 28.