Automating drawing of polygons (square) around number of points in QGIS?

QGIS ships with OGR which has a method GetEnvelope():

# Returns the geometry's extent as a list (minx, maxx, miny, maxy)
<geom>.GetEnvelope()  

You can easily use it from within your QGIS python console. If you want to convert those coordinates into a shape? Have a look at my answer to this question for some info on modules which will help you turning this list into a vector file. There you find also further details and some helpful links.

You could use the mmqgis plugin for QGIS, which provides some more user-friendly implementations and interfaces of some QGIS functions. It adds an additional menu button next to "Web" and "Processing" from where you can choose "Create > Create Buffers" to get this interface: You get the option to choose in what units you want to measure the radius, how many edges you want - choose 4 for a square - and how the buffer should be rotated. At least in my case I also needed to rotate it by 45 degrees, otherwise it would have had kind of a rhombus-orientation. I did not try out the attribute option.