Here are 3 inexpensive squares commonly available in many hardware/big box stores. A Sawnson 6", an Empire 12" and 16".    

Using the line test, I see how square they actually are. The Swanson 6" doesn't come close to passing the test. The difference between the 2 lines is how out of square it is, Given it's so short, this is pretty bad    

The Empire 12" is pretty good. This one passed. But - the head is aluminum and the rule is stainless steel. Over time the steel will wear down the aluminum and it will be off

The Empire 16" is way off. I am going to correct this one following the steps in this video

These are the squares I actually use most often. They are accurate and easy to true when need be

All you need to make a square is 2 small pieces of hardwood. The harder the better for the "rule". Here I have some bubinga and some walnut. The bubinga is just over 1/4" thick, and the walnut is about an inch thick. 

I need the walnut to be 3/4" thick, so I use a marking gauge to mark the 3/4" depth all the way around

Once that is done I plane it down to the lines

Now I need to rip this down a bit. Here I am marking about 1 1/4" all the way around like the previous step

Then I toss it in the vise and rip it with my ryoba. Be careful here and try to cut as straight as possible

I was able to stay on my line when ripping so all I needed to do was remove the saw marks with a plane. If I was a little off I would have made a gauge line and planed to it like in the previous steps

I then shoot the edges straigt

I want to see how wide the blade will be so I need to bring it in parallel first. I use the marking gauge to reference one side to the other

Then I plane to the gauge line

Now that the bubinga is a consistent width I can proceed with cutting the bridle joint.  

. I take my 1/4" mortise chisel and place it approximately in the center of the edge to make a mark

I then set one marking gauge to the depth. I want the blade to be about 1/4" proud. I take another marking gauge and align it to the edge of the mark, then transfer it across the the face and the top edge of the board. I then do the same to the other side. If you got lucky with your chisel mark dead center and got the piece to exactly 3/4", it will already line up

After making the gauge lines, I CAREFULLY saw down the board. Be sure to go slow and stay inside the lines. A little paring will be necessary, but accurate sawing will save a ton of time later on

This side is looking good, now onto the next. You'll notice at the bottom I established the baseline with a chisel cut first. This helps prevent oversawing

Now the other side it cut

I take out the waste with a mortise chisel being careful to keep the inside shoulder straight. Once I get halfway through, I filp it over

Doing the same from the other side, the waste will pop right out

I put it in the vise and pare the floor and the walls square

Now that it's clean it time to test fit. It's too tight, which I expected. I want a very tight fit, but I don't want to split the walnut

It's far easier to plane the bubinga than to pare away the walnut evenly. I take a few strokes with a smoother then check until it fits

Next I trim a little length off the blade and square up one edge 

Then I cut an arbitrary angle on the end

One reason I love using Japanese saws is they cut so smooth. I don't even need to plane off the saw marks. I cut the angle a tad bit away from the edge so there is no sharp point

Now to add some details. I like to chamfer the outside edges with a chisel so they don't split

After that I plane a chamfer on the long edges with a block plane

and then chisel a strong chamfer on the bottom with a chisel 

Now to test.
This is the pencil test I was talking about before. With the square against a known flat surface - like the edge of this plywood, I draw a pencil line

Then I flip the square over and draw another pencil line right over the current one

It's off. Not a bid deal. because it goes to the left, the back of the mortise/bridle needs to be pared down a bit.  I adjust and test until it's good. 

Now I add some glue and clamp it. I let this sit for about 24 hours to ensure everything has ample time to cure. After I take the clamps off, I will check it again and make any adjustments needed by planing the blade. 

After The glue dries, I like to remove the end piece with a flush cut saw and then plane it flat

I found a left over bubinga plug I had cut a while back but never popped out, so I added that in

Then scraped it flush in prep for a quick finish 

For the finish I chose to use shellac. I like to pore fill walnut and bubinga. It makes them look much smoother. After a light coat of shellac, I rub in some fine pumice with a cloth slightly dampened with a very light cut of shellac to make a slurry that fills the pores. Then once dry, I pad on more shellac

All done and ready for use