ETA: This answer has the wrong version & story format for the question, my bad. It might still be useful to people using Twine 2 and Harlowe 1.2.4.
You can make a link that changes two variables like this. First, set the variables.
(set: $firstvariable to 1)
(set: $secondvariable to false)
Next, make your link where changes happen.
(link: "Soon your variables will change.")[(set: $firstvariable to it +1)
(set: $secondvariable to true)
(goto: "Passage where variable are different.")]
Admittedly this is a link not a button. I don't know how to make buttons. But I hope this helps.
I learned it all here -> https://twine2.neocities.org/