A pixel is essentially the same thing as a single point in Photoshop. If you zoom in far enough to see the pixel grid, and select a brush with a square brush preset set to 1 pixel, then although you appear to be able to hover between pixels, that is an illusion.
As you move the cursor — with the Info panel close to the cursor — the coordinates will snap to the nearest whole pixel. If you try to stamp down with your 1 pixel brush, it will always be placed exactly on the nearest whole pixel. What else can it be?
With lighter tones it can be difficult to see the pixel grid. You can use the same work around that we use for work paths... Place a layer filled with black at the top of the stack, and set its opacity to 50% (or there abouts)

Incidentally, there is nothing the regular posters to this forum like more than a question they don't know the answer to, because we can then try _very_ hard to work out the answer. There's kudos to be the first with the answer (unspoken but I am sure we are quietly pleased). But most importantly, we get to know a little bit more about the program we love. You'll not find another group of people who will try harder to answer even the most obscure and difficult questions.
[EDIT] I meant to say that if you copy your selected line of pixels to a new layer, then guides will snap top the end of that line when dragged out from the rulers. Make sure your rulers are set to Pixels, and the job becomes super easy.