CC Weedcontrol ,if your quarter lines align from north to south and from west to east, there is a solution for your issue.
the points are align horizontal and vertical .and that is the clue
the solution is :
Frist : it is attribute task . you will add x and y coordinates field Add XY Coordinates—Help | ArcGIS Desktop
so you can classify each align points into classes , you will define the tolerance and group each align points horizontal into class and vertical points into class( the point has two classes one for vertical and other for horizontal)
Second :you will use Sort Tool Sort—Help | ArcGIS Desktop and you will add two fields for sort ( first horizontal class field , second x field)
and use Sort tool again for vertical and y fields.
Third : now you can get your quarter lines if you use Point To Line Tool Points To Line—Help | ArcGIS Desktop and use ID field of sort in this tool. And class field for line field in this tool.
I hope my answer be clear.