Ok, here is my method. It only takes a minute or two once you get the steps down correctly. This was done on a 24" dia. sphere with 32 segments.
1# Cut the sphere square so that there are four triangles.
2# Select each cut and use the connect command set to one to form a square. Adjust the size of the square as needed with scale. I also used edge constraint(under edit geometry) for all this to make sure it stays flat.
3# Select the four faces within the square you created and inset by .05 inches. Do this again to from three squares within each other as shown.Now, cut from the corners of the inner most square out to the edge of the sphere square as shown in image 2.
5# Select the inner four triangle faces and then hit grow one time to select the faces surrounding them as shown in image 2.
6# Extrude the faces inward -.05 inches and close the dialog box.
Extrude again the depth you want the square minus .05 inches and close the dialog box.I did 1/2" for this.
Extrude again -.05 inches and close the dialog box.
(NOTE) Be sure to use Local Normal when extruding.
7# IF you want to, you can collapse(weld) the 2 vertices for each outer edges and move them to the center of each outer edge of the square.It’s not neccassary, but it looks cool.
Thats it! You should not have to adjust anything to get a smooth square hole.