Let me suggest a "Japanese" style pull saw. These are available in many sizes and are able to do both plunge cuts into your workpiece as well as cut right into a square corner.
Something like this:
These are on the high-end but you can find similar saws at most home centers and hardware stores. They are very useful!
You're not going to be able to reasonably cut the square out with your hand saw. You'd have to scrape through the wood instead of cutting it.
Pick up or rent a coping saw and a hand drill (See saw below). Drill a hole in the four corners, remove the blade from the saw and run it through a hole, hook it back up and saw away.
I am not sure if there is an absolute requirement for the handrail to be fully continuous the whole distance. But you may actually get a better looking installation to have the hand rail be mounted in two separate pieces that simply meet near each other.
Typically it is done with a miter cut to get to level, or up easement, then a 45 degree level turn , then another miter cut to resume the angle or over easement, this will get the job done. You can do it with all on-site miters, or you can purchase "fixtures"... parts that are "pre-bent" to make smooth turns. Where I have "up ...
Many moons ago, last century, I used and appreciated a program known as winmiter. It no longer works on today's technological magic, but there's an online free alternative known as Pipe Joint Template with an option to purchase and download an offline version with more features.
The program allows you to print a paper template to mark the cuts necessary to ...
I've always cut cardboard tubing (for concrete forms) with a circular saw. You could also use a table saw and roll the tube on the blade with the end against the fence for crosswise cuts.
You may have to experiment with blade tooth type to minimize fuzzing and tearout. I'd start with a thin-kerf framing blade.
Search for "fishmouth joints" which you'll find in bicycle frames, tubular hot rod frames, and light aircraft frames, among others. Websites catering to those pursuits will likely have information and even patterns or better yet pattern-generating programs to cut the "fishmouth" shape into the tubing.
The other approach is to have a ...
The blade looks bent. I've drawn lines along the spine of the blade as shown in the photo in the question. If we assume the background is flat, the point where my red lines cross isn't sitting on the surface, and is bent towards the camera. See also the shape of the shadow, and the non-round holes. On a non-flat background, it's possible the blade has ...
I quote from an online article concerning saws of all kinds.
A jigsaw is a handheld powered saw. It has a smaller blade and finer
teeth than a reciprocating saw. It moves vertically. You can change
the speed to make it go faster or slower. This saw is designed to cut
curves and other non-straight lines.
A jigsaw is fully capable of making a decent cut. Will it be as nice as a table saw or as a pull saw? No. But the cut you have here doesn't appear to have any straight lines at all.
High blade speed
Don't advance the blade into the wood faster than the blade wants to cut the wood
With those three things I don't think you even ...
Ever considered why it has its name? It's designed to do just about anything except cut in a straight line!
Its blade is thin, front to back, on purpose. So it can and will go round corners easily. To cut straight lines, the blade needs to be long, front to back, so it's difficult to turn. As in tenon saws, which even have a strengthener to stop them flexing,...
As a novice-level wood cutter, I will say that what helped me the most with learning how to make straight cuts using a jigsaw was to slow down. You can certainly cut in a straight line, without a guide, if you take it slow, with some practice. More pressure against the blade equals more likelihood of not being straight - let the blade do the cutting, not ...
Forget the guide block. Seriously.
Jigsaw blades wander for a variety of reasons, and all of them will cause you to have a non-square cut, even if you have a square guide block.
Instead, practice cutting to a line. (Or more accurately, cut most of the line on the waste side.) If the 'top' of the board is supposed to be the nice side, then mark the line on ...
Jig saws are designed for cutting curves.
You can cut straight lines and generally you followed the correct procedure - using a straight edge. Your picture of the cut is pretty blurry, so it's hard to tell for sure, but there are several possibilities:
The piece of wood you used as a straight edge isn't as straight as you thought. Usually one uses a metal ...
A jigsaw isn't designed to make perfect cuts. Straight cuts can be made but the stock has to be clamped down and the fence, guide block, must be clamped down too. You can't hold down a guide block with one hand and operate the saw with the other, the guide block will move and the saw will jump, as is obvious from your pictures. Clamp everything and use both ...