Hmm, and because of that I have to replace the whole A/C and heating box?
I did some research and I think I can try spraying some WD40 with straw around the edges of button so that the WD40 can penetrate into it and may unstick button.
I was looking for that kind of advice.
All that means nothing if it simply gets stuck again.
And a tip of the type OP might be looking for............WD40 is one of the worst things to use longterm, when it dries up the residue is best described as sticky bubblegum and will gum up anything it got sprayed into. So, sh-tty advice.
OP is the one who said button does not come back out, self limitation, good he didn't listen to himself there.