We have those big blocks here also in the old places. My house has a field stone foundation up to grade and then they switched to the blocks you mentioned. I have always called them sandstone and they must have had some big guys back then setting those monsters.
So the house doesnt have a basement and the blocks are only a couple feet below grade so you are not in a deep frost line location. I assume you will be working from outside by enlarging the hole the dog started.
The blocks should be brought back up and then the easiest way to support them would be to pour concrete below them. Without photos and just guessing sounds like get below and jack things back up and brace it with something stones bricks etc. over a small area. Get the jack out and pour the rest.