Let's see if this helps explain it.
First, you are absolutely correct that the begining and end segments of the path need to be at 90Β° to each other for a 1/4 sector and for Follow Me to work correctly. Instead of adding line segments to your 90Β° arc, though try the following.
http://farm5.static.flickr.com/4154/5190512797_5e0876427d.jpg
In this image, I've added red lines over the line segments at 90Β° to each other. Notice these are half the length of the other line segments.
Instead of drawing an arc for the Follow Me profile, draw a circle and make sure you drag it out on axis. Then rotate the circle on its center by half the angle between vertices. In the case of a default 24-segment circle, that would be 7.5Β°. This will set sides parallel to the axis lines. Divide the circle into quarters or as I did eighths and discard all but one of those. Now you have your path for Follow Me set up correctly. The end segments are at 90Β° to each other and when you put two sectors together, they meet on a line rather than at a point. the other benefit of this is that when you hide seam lines, they are actually hidden. If the seam line occurs on the vertex of the circle, the edges won't hide nicely and softening then will do strange things to the appearance of the faces.
It takes longer to read this than do it.