Can a shapefile have only a single point?

No, you cannot use a shapefile that has only a single point.

The extent of a feature dataset or feature class in a geodatabase is defined by the allowed extent of the coordinate system assigned to the data. In a shapefile, the extent is defined by the extent of the features in the file - points, lines, polygons, or multipatch. Since a single point does not have an extent, a shapefile that only contains one point may not display, or may cause other issues in the software. This is not a bug in the software. This is a limitation of the geometry display in a shapefile data model.