Tab Auto Sync syncs in real-time your open tabs in Chrome (as long as you are signed in - which you currently are):
Real-time synchronization of tabs and windows between computers.
Closing chrome will save your tabs. To remove tabs, close the tabs. If you close the last tab, chrome will close and next time you will be shown the new tab page.
In my test, this worked exactly as you want, you don't even need to use a button - you just close your tabs on one computer, they open on the other (when you open Chrome).