The post office closed at Boston (on Valley Road) in 1879 and moved to Moline that same year.
Thomas E. Thompson has always said that there was absolutely nothing left of Boston in 1884 when he and his wife drove down the former Main Street in a buckboard. All the buildings were gone.
So, at the end of the century, perhaps, there was just a neighborhood area there like Busby or Belknap.
Boston, Kan., Jan 9 [1896].—Unknown men called at the house of Edward Welch last night, blindfolded him, cut off his left hand with an axe, and robbed him. The community is aroused by the outrage and bloodhounds were put on the trail of his assassin, who took away the severed hand.
New York Times
There was another Boston, Kansas, in the state before our Boston came into being in 1871. The other Boston changed its name—to Manhattan.