I'm certain it is possible to interface those buttons with Arduino, but it would probably require opening them and making some simple, nondestructive modifications.
The company may not be interested to give you the required information, but you could try telling them that there would be additional buyers if the info were published.
If they aren't forthcoming, it would not be difficult for someone with experience to reverse engineer enough of the circuit to get something working with Arduino.
On the other hand, you can just buy heavy duty buttons of that size, that are just buttons. Attach them to an Arduino and away you go! (An Arduino can be programmed to say words upon button push).
Thank you for the reply! Me and some others are part of a testing group for them. I've talked to the person that developed the buttons and I'm sure he could explain some inner workings if needed. it will ofc remove the guarantee if anything breaks. Obviously.
I'm ok with opening them up and adding nondestructive modifiations. I would like to continue using these buttons because my dog is used to the size/shape/sound/location etc. It's would be a headache to teach her again. We are also part of a research on how to teach dogs. Changing buttons would affect the data a lot.
The recording of the buttons in a log would also be great for research. That's my primary goal. Collect data.