On the picture below you can see a vertical line going pass the horizontal line for a bit. How can I meet the end of each line to make a perfect even square rather than showing one higher, lower, or further to the left, or furter to the right positioning?
The anchor points need to be in exactly the same position.
As I (mis)understand it, to obtain it you will have to (deselect and then) move the lower right path(s)/Group as a whole by (first) ClickDragging the relevant Anchor Pont (possibly followed by other movements), to avoid distortion.
Exactly how and which Anchor Point depends on the actual path/Group structure.
In any case Smart Guides are your friend, helping you to snap in the right way.
The easiest cases are if you have either:
A) Fill/nostroke paths, in which case you can just ClickDrag the top right Anchor Point to snap to the corresponding Anchor Point of the upper left path(s) Group,
B) Stroked paths with Projecting Cap, in which case you can just ClickDrag the top end Anchor Point to snap to the corresponding rightmost end Anchor Point of the upper left path(s) Group.
More is required if you have:
C) Stroked paths with Butt Cap, in which case you can first:
C1) ClickDrag the top end Anchor Point to snap to the corresponding rightmost end Anchor Point of the upper left path(s) Group;
C2) Move the path(s)/Group up and to the left by half the Stroke Weight; you can do that with Object>Transform>Move or by subtraction (and possibly addition depending on version/settings) in the X and Y boxes of the Transform panel.
Turn on snap to point (possibly turn off snap to grid, snap to pixel, snap to glyph depending on your set-up and other items around).
You can also click on a point and specify it's exact position.