A word to describe the subject of an action

Solution 1:

The answer to this question will depend on the syntax of the phrase into which you wish the answer to be injected.

The button has a property called pushed

make sense in programming contexts, for example, where the property's datatype would be a boolean (if the button is two-state, either on or off).

This programming example is based on the linguistic fact that the past participle ("pushed") can be used adjectivally to convey the notion of the 'state' of something.

If we abandon the syntax in your fill-in-the-blank example, we could rephrase the statement and say "The button is pushable" or "The button can be pushed".

One of the button's attributes is that it can be pushed.

Strictly speaking, (i.e. in natural language not programming terminology) "the button has a property called pushability". The word "called" wants a noun there, not a past-participle adjective reflecting state which in some circles (e.g. programming) is used like a noun. The naming conventions of programming are sometimes unnatural.

Solution 2:

The button has a triggerable or triggerable behavior called "push"? The button receives "push" actions? The button has capabilities of reception of "push" actions?

To use more programmatic terms, The button has listeners (or handlers, etc) for "push" actions?

If you are trying to use this in a programming context, you might want to check how it's phrased in the documentation of the language you are coding in. You might even consider asking this question in Stackoverflow.