I did encounter one seemingly endless loading screen, but after restarting the game I personally managed to play several games of Conquest, Rush and Conquest Large without issue. Joystiq's Earnest Cavalli, who has lost a total of four save files, managed to play three Conquest matches without any crashes or corrupted saves.
Playing the Xbox One version earlier today, I experienced one crash but was otherwise able to play Rush, Conquest and Conquest Large with no significant problems. There was one moment when the Kinect decided to interpret some in-game sound as "Xbox, go home," thus disconnecting me from my session, but that's hardly Battlefield's fault.
The server browser is still experiencing problems on both Xbox One and PS4, specifically that it can't seem to locate Conquest matches at all. Applying a filter to locate Conquest matches simply returns no results, regardless of any other restrictions placed on the filter. For now, if you want to play Conquest, it looks like you're stuck using the Quick Match option. Other game types, including Conquest Large, show up in the server browser without any problems.
Let us know how your PS4 and Xbox One Battlefield 4 experience has been in the comments.