As timberland said, it is possible to have a solar eclipse on the soltace, the problem is when this is going to take place. It is very easy to calculate when and where each solar eclipse will take place.
Specifying a solar eclipse on a certain date in a certain location is going to nail down a certain year... I checked on the NASA table, and I didn't find one that will happen in the next 100 years. I did find an eclipse that happened on Dec 21 in 1805, but it was only visible in the Southern hemisphere.
You can look for yourself on the
NASA calculated table of Solar Eclipses.
Other scientific ways to make the darkest day for your film are a volcanic eruption (I think this is the real cause of the darkest day in history) or giant sunspots (not really realistic, but probably acceptable for a science fiction story). The side effects of these ideas might be too great for your story though.
good luck.
Good luck.