Starting with a spotty raster like this, where 1 is a black square and 0 is elsewhere:
Use "Polygonize", and then deleting the code=0 polygons gives me this:
which is all the pixels outlined as polygons. If adjacent pixels are 1, then the whole block is outlined.
Then use "Centroids" to make a point at the centre of each polygon:
Note how there's one point assigned that that block on the upper left which was two pixels (sorry things don't line up exactly).
If I overlay the points and the original raster you can see this more clearly:
If that's what you want then job done, I can't see a direct way of turning every pixel into a point, but I'm sure it can be done if that's what you actually want.
Once you converted to points, you can style them and they can be a large and constant symbol size. You can even make them square symbols if you want, but they'll be centred on your original pixels.
If you want every pixel to have its own point, then save the raster in XYZ format, which results in a text file of X,Y values for every grid cell with an extra column of the pixel value. This can then be read in as a vector points delimited text data file.
for more, although recent versions of QGIS look a bit different. The crux is saving as a file with a
.xyz extension so that GDAL uses the
XYZ output driver.