In Twin, the answer looks like this (that is, this is the usual text that you need to click):
And I want this text to be something like this printed on the button you want to press:
Is it possible to do this?
Yes. Instead of doing:
[[Link text|Passage name]]
<<button [[Link text|Passage name]]>><</button>>
See the <<button>> macro for details.
If you want to change how the button looks, like different background or font colors, then you'd just need to change the CSS for the buttons. See here for a CSS tutorial.
Hope that helps! :-)