If you want a precise 256x256 pixel raster file containing a 1 pixel checkerboard pattern, you should definitely use a raster program (or program the exact output you want from a programing language).
For example, if you use GIMP (which, as Inkscape is Free both as in Beer and as in Speech), you can simply create a new 256x256 pixel image, and select the Filters->Render->Pattern->Checkerboard option to get your desired image, and file->export to get your output exactly as you want it.
(As an extra note, don't try to save such an image as a JPG file -
you'd be losing the pixel precision again. Use PNG or even GIF)
As for your question (I've described how to deal with your problem) -
it is the very nature of Vector images not having a "one to one" mapping
to pixel images. That is their idea: to allow a geometric description
of a scene, instead of having Pixels. With that, they do allow for
much more precise arbitrary scale images, using much less computing
resources (mainy memory usage). For example, a 3000x2000 pixel image
containing a single smooth red circle in it's center could consume
up to 18 Megabytes in an uncompressed form - that is roughly equivalent to the textual information contained in 5 full Bibles. (although most file formats use compression
that could drastically cut on that). In contrast, the same image described in a vector format which could have the very same look would use at most 1Kb - that is
18000 times less resource consuming - exactly because you don't need
to retain the information for each pixel.
In the case described you are trying instead to define one vector object for each pixel.
Not only you don't need Vector images re-scaling capabilities, as you are doing the exact opposite of what vector images where first thought for: since each pixel uses
3 bytes of information, and describing a square in a text-based vector format
lieke SVG will take about 40 bytes, you are consuming 10 times more resources than you should with this image, instead of sparing any. And moreover, as you found out, subject to rounding errors due to anti-aliasing.