Are you sure she hadn't already started giving birth? They usually do it in the early morning. Also if the fry get eaten it is hard to tell if it has happened. Normally when they are squared off it means they are currently having fry or are about to within minutes or hours at the most. She may only have one every 30 minutes or so, so if you see her in between birthing it may appear as if she is not giving birth.
The thickness observed "under her gills" is actually her stomach, indicating that she has just eaten a lot. The fry are carried farther back by her gravid spot.