Point of clarification - if I understand correctly, you have all the lines depicted in the first example, but need to cull them down to only the quarter section ones (so a subset) - is that correct? If that is the case, are the lines already attributed so you know which ones are the section lines (so there is a basis to then do the quarter-sections)?
Also, points are shown in the examples. Are the points attributed in any way that would help define which lines are section lines and which ones are quarter-sections? I ask as one way to do this might be to build the lines from the points using Points to Lines, but before doing so the points will need attributes that allow the correct assembly.
Points To Line—Help | ArcGIS for Desktop
Another question - are the lines all roughly North-South and East-West, or are some of them in other orientations (like NW-SE)? If they are all North-South and East-West, one may be able use their increasing X or Y values to help in differentiating them.
Chris Donohue, GISP