I must be the exception to the rule because I like making half square triangles. I make my fabric squares 1" larger than I want my finished square to be and draw a line corner to corner and sew 1/4" from either side of the line. It doesn't make any difference when you cut them apart if you are a little off because they are already stitched. You can even be a little off on your sewing too because you're going to trim the block.
I have two methods of squaring up the block depending which ruler I find first. Eleanor Burns has a ruler for squaring up HST that you don't press the square open but trim first. You put the corresponding size line on the stitching line and cut the two raw edge sides to the right size and then press open. The other method is to press the square open and then take a square ruler and put the diagonal line on the ruler along the seam line and trim two sides. Flip the square around, put the just trimmed sides on the right measurement of the ruler again with the diagonal line along the seam line and trim the remaining two sides. Either method gives me perfect HST with no bias edges. My avatar is almost all HST.
As far as chain piecing that can be done too but you need to overlap the points of the squares as you are feeding them into the machine. If you line up the lines you have just drawn as you are feeding the next square into the machine you'll be in the right position to sew. There will be some gap between the two stitching points because your presser foot will dictate how far you can overlap the two squares. As you become more proficient with it you will know when to feed the next square in and greatly reduce the thread gap between the two. It really is quite a bit before the end of the stitching of your preceeding square and there is quite a bit of overlap between the two squares. I chain piece my HST this way.