Thanks for the report; is there anything unusual about your setup with respect to Sync, such as trying to use a self-hosted sync server or working behind a corporate firewall?
I've never seen this before so it's hard to know how to diagnose, but some ideas include...
clicking the Sign in to Sync button, on page with 'Sign - in' button, no matter how many times I click, it does not work.
If you open the browser console while clicking on these buttons, does it show any errors or other hints about what might be going wrong?
You could also follow these instructions to see if there are any error logs generated by Sync, and attach them to this bug if so.
When I reinstalled and got a different login page, when I enter my credentials it gets stuck on loading and never finishes.
On the page where you enter your credentials, please try opening the web console and see if there are any errors logged.