For quick and dirty, you could just use a spherical approximation.
For lat and lon in degrees, placing lat1,lon1 SouthEast of lat0,lon0 per your example:
dlon = RadianToDegree * BoxSizeMi/EarthRadiusMi
dlat = dLon / cos(lat0 / RadianToDegree)
lat1 = lat0 + dlat
lon1 = lon0 - dlon
For a box size on the order of a mile, this should give a visual result indistinguishable from a more accurate calculation (say if you were plotting the box as lines on an image while somebody moved the mouse pointer). Pick a reasonable earth radius in the same units as your box size; the geometric mean of the equatorial and polar radii would be one simple choice.
You can get far more accurate, but in true high accuracy the question is ambiguous; for example going east exactly one mile and then south exactly one mile is not the same point as going south and then east (because the meridians or north/south lines are not parallel except right at the equator).
There are intermediate precision solutions if the above is too dirty.
Things get wacky near the poles for this reason, whether you attempt precision or use quick and dirty approximations.