Hi!
Not sure about THREE.LineSegments() with THREE.Geometry(), but, if you work with THREE.BufferGeometry(), then .intersectObject() method returns an array of objects and one of properties of those objects is .index, that represents the index of the neares vertex of the intersected line segment (if I got it correctly, though the documentation doesn’t say about it).
Thus, when you know the index of one point of a line segment, you can find the index of the other point just like that:
suppose idx is the index you found by intersection
let idxNear = idx % 2 === 0 ? idx + 1: idx - 1;
So, you know indices of vertices, you can obtain their coordinates from line.geometry.attributes.position with .fromBufferAttribute() method of THREE.Vector3().
Just a related fiddle, where you can see how to find the indices of points in a line segment:
https://jsfiddle.net/prisoner849/hu6jjj4p/
One more example of interaction with points (an indexed THREE.BufferGeometry()):
http://jsfiddle.net/prisoner849/go0dfwo5/