Ensure your link is using UTM parameters
The best way to identify users landing on your site is to tag the incoming URL with UTM parameters using the Google Analytics URL builder. The URLs built with this tool can be long, so when you share them on Twitter, you will want to use a URL shortener.
Visitors that enter your site via links with UTM parameters will have their traffic source tracked throughout their session.
Create a segment for this traffic source
Click on "+Add Segment" on a Google Analytics report. Then click on "Traffic Sources" and enter the info that matches the UTM parameters you want to track.
Save the segment and make sure it is the only one applied.
At this point all your reports in Google Analytics will only show data for users that came through this link to your site.
Make clicking the button into a Goal
When the user clicks the button they may visit a specific URL on your site, or you may have code set up to send and "Event" to Google Analytics. You can then make a "Goal" based on pressing the button.
View a report with the info you want
You have a couple options. The "Behavior" -> "Behavior flow" report will show you the paths that these users took through the site.
The "Conversions" -> "Goals" reports will show you information about how many users met your goal and which path they took to get there.