I was having the same issue, where I would be signed into xbox live, be able to purchase, get achievements and etc..., and was getting that same cannot connect to EA error.
When I went to this site: http://starwars.ea.com/starwars/battlefront - I could see my xbox live gamertag and see that it was the "active account", so confirm you see the same thing here first.
After that, I am embarrassed to say, all I did to be able to connect was to reset my stupid router while signed into my xbox one, on the title screen for SW Battle Front.
Immediately after resetting the router, which again was letting me connect to xbox live and so on, I connected the first time after pressing Y and have been playing all morning.
Hope this helps, and sorry EA is utterly useless in providing answers to any of us!
