If you can redefine your square section to have more segments, it gives you way more control than adding in blend vertices in the sweep ui:
Still not quite sure how you can make such a part:
Here are a few options.
in 1A, I redefined the sketch of the square, much like Paul did, but I added the split to the verticals instead. I then dragged the white dot (circled) for the start point for one curve around to better line things up with the other to eliminate the twist.
In 1b, I simply rotated the white dot on your swept blend, no additional points.
In bb, I created a datum point at the end of your trajectory sketch, lengthened that sketch and then created the transition as a boundary blend and surface cut instead. It creates a nice, even funnel shape, but where it intersects the square hole isn't planar, however.
OK, I did one more, incase having 6 even splits and a planar intersection with the square hole is important. It's a bit more involved.
First, you have to redefine the square to put vertexes at the 60 degree marks. Split the edges at the centerlines and then, at each corner, use operations > convert to > split on the "L" shapes at the corners (highlighted) to a single entity:
Then I needed to re-select the square sketch and right click on the extra vertexes and remove them. The reslut:
Hello Paul Sobejko,
please, mark correct answer if your question has been answered. It can help others users in future.
Note:
If sketch control points are not conected (surface is twisted) als you wish and everything looks right, than try to check sketcher orientation reference.
It should be the same for all sketches, that are used in swept blend feater.
Swept blend is "kind of feater" where sketcher orientation reference can went you mad 
. It doesn´t happen often, but it happens sometimes.
