We were able to narrow the problem down to the UAG (Unified Access Gateway) and there has been an update (SP4) that supposedly addresses this issue: http://blogs.msdn.com/b/testingspot/archive/2013/11/27/forefront-unified-access-gateway-2010-sp4-released.aspx
Until the UAG is updated, here is a work-around:
1) Launch Internet Explorer and go to tools menu and compatibility view settings. Add the site to compatibility view
2) Hit the F12 key to open the Developer tools on the bottom of the screen
3) Change the user agent option to Internet Explorer 9 or 10
Now you should be able to browse back to the site and login with your credentials.