Is there an option to change a default camera app so when I press the button it would open other program than default? I'm using Camera360 app, if it matters.
Now a days is not possible for every app show as an option to be the default camera app.
To change the default camera app you need to go to Settings, applications, photos+camera and select the camera app of your choice in "Pressing the camera button opens".
I have the app you had mention and it is not listed as an option.
I suppose this can be a restricted permission allowed only to OEM but can't confirm.
If you go into the Settings app, scroll across to the applications tab, you should find a "photos+camera" section - in this, there is an option to choose the camera application to launch when pressing the camera button.