I finally removed the counters from the table today. This would have much easier had I know how they are supposed to be attached, and if they had been attached properly.
The correct screw is a 10-24, flat head cap screw, that takes a 1/8" Allen wrench. The left one was like that, the other had a couple of button-head screws. If the correct ones are there, you can reach through the holes in the top of the ball storage box with an Allen wrench and take the cap screws out, and then lift the counters out.
Once out, I saw that some of the counter wheels were stuck to the axle. There was only one felt washer on the two counters; there should have been four each.
I had to punch the axle out. The problem was that the axle had rusted, and the increased diameter, due to the rust, caused the wheels to bind.