I think John in Kansas is exactly right. The solstice isn't a day but a moment. Every spot on earth experiences the solstice simultaneously, but according to local time, half the earth experiences it in the a.m. and half in the p.m. If it falls in the a.m., I would expect the night before the day of the 21st (assuming the solstice falls on the 21st) would be the shortest night. If it falls in the p.m., I would expect the night after the day of the 21st would be the shortest night.

Here's another bit of trivia: The latest sunrise and the earliest sunset do*not* occur on the same day. One (I forget which) occurs a few days before the solstice and the other a few days after. I have the vague notion that this is because the earth's orbit around the sun is not circular but elliptical, but I would have a hard time explaining it.