Show/Hide a page until I click a button

You can use template pages for this.

Not directly, but you can do it with a trick. Draw a large white button over the the page and fields you want to hide.

Then on the button that the user clicks, add an action to show/hide (hide) the large white button to uncover the hidden fields.

You can't create or edit forms in Acrobat Reader where you posted, though, you'll need the paid version.