I found a fairly long loosing streak. Tunica County MS (1948-1968)
That may be the only Thurmond-Stevenson-Stevenson-Nixon-Goldwater-Wallace county. It's probably one of the only Stevenson-Stevenson-Nixon-Goldwater counties, too. Most of Nixon's counties in the South in 1960 voted for Eisenhower at least once.
Actually, that County voted for Humphrey.
Walpole, Wilmington, and North Reading are Massachusetts Townships that last voted for a winner in 1996 at the latest, and possibly earlier (I lack access to data for New England Township results prior to 2000). Although they are not full fledged Counties.
A total of 40 Counties voted against both George W Bush and Barack Obama twice each. All have since become solidly Republican (you can find the full list in an October 2016 Sabato Crystal Ball article).
Charles County MD, and Orange County FL voted against both Bill Clinton and George W Bush twice each. Both have since become solidly Democratic.
Also, I believe that it is cheating to include Counties as losing streak Counties if they voted 3rd party sometime in between.
