The Playstation 4 does not need an internet connection, however the day 1 patch does enable a wide variety of features. If all you want it for is to play games though, it should work entirely internet free I'm fairly sure
Games should come with the system update version they require to play, if any, a la the PS3 I believe, but I haven't encountered this yet since the PS4 updates it self...on the internet, anyway. You should also note a few games require day one patches either for stability or performance tweaks which could be a deal breaker. A couple games require patches for higher resolution, and BF4 at the least requires quite a few patches (but it's not much of an offline game anyway).
The Xbox One needs a patch to be used offline and will be basically dead with no connection or patch of any sort. Again, not sure if you can update the system's OS using game disks like you could with PS3, if someone knows whether you can that'd be great to know. Day one patches are at least as much of an issue on Xbox One though, for example I think Forza adds a notable amount of content with a patch including, I believe, the ability to finish career mode.
The simple answer is, yes. The Xbox One has many features that require more bandwidth such as video and music streaming (as most content is served in higher resolution). However, these services are not required to game.
Services such as chat might require higher bandwidth but only marginally as Codec's, what is used to encode and decode the voice before sending it, allows for higher quality sound to be compressed into less data.
By your description I believe that you do not fully grasp the reason why you are lagging out. Indeed, lag is linked to bandwidth but it is usually not the deciding factor. What is more important is latency- Please check a site such as Speedtest.net to see what your latency is.
Latency is the round trip time of data you send out. Basically, imagine it as a postal service. Latency is the time required for a letter you send to be responded to, i.e, when you hold the reply in your hands and can read the response. This is a very important concept as online servers rely on the responsiveness on all participants. So when the host server detects that you take to long to respond to the packets it sends your way, it will drop your connection to avoid impact on the quality of the game for other players. That is why you lag out.
Therefore, a low latency is what you want. However, latency is only loosely tied to bandwidth. It has to do with the quality of your equipment, if you are on wireless or wired, your ISP, your geographical location, and the location of the remote server. There are more factors that are basically related to the above. So when you play on a server hosted on a different continent, your latency will be higher. That is normal and expected.
Now, sudden drops in latency are usually what cause sudden lag or drop-outs. They are caused by bursts of data that clog up the "pipe". Again, imagine it as a postal service. While on a normal day they can receive and forward your mail in a few hours, during Christmas season they are swamped and will take longer. This is to be expected. Similarly, if you are saturating your connection or equipment with downloads, streams, or similar, your equipment, especially older equipment, might struggle to forward all packets in time and connections might drop.
The tricky part is that there is not always something you can do. If you live in an apartment building that shares a physical line, the other tenants might have some activity that is transparent for you but still impacts your connection. The same might hold for your street as all connections on a street usually share the same physical line which uses the same physical equipment down the line.
One way to go might be to contact your ISP and ask what you can do. They should be able to help you. Personally, I'd advise you look into getting on a connection based on optical fibre. They feature less latency and can carry more bandwidth while alleviating some of the problems I mentioned above. If you want you can take a speed test and post it as a comment, I'll try to give you more streamlined advice.
Best Answer
From Microsoft's official statement post-the E3 uproar, back on the 19th of June:
(their emphasis)
That "one-time system setup" includes an initial day one update of about 500MB that the console needs to download to be fully functional, but you don't need to sign in to your console or Xbox Live to receive that update, the console will find and download it as soon as it is first internet connected. (source: this was the first thing my Xbox One did yesterday after switching on and checking my wired internet connection, and before taking me through the initial set up of signing in and calibrating the Kinect).
Of course that statement was published under the name of Don Mattrick, then President of Microsoft's Interactive Entertainment Business, who has since left Microsoft to join Zynga, so I can understand anyone worrying if that statement is still true.
Marc Whitten, who is still, Microsoft's Xbox One chief product officer was interviewed by Kotaku, and this is what was said:
I don't believe that there has been anything since contradicting that statement.
(Addendum: it was briefly possible to pull down the day one patch on a PC and transfer to an offline Xbox on a USB stick, but Microsoft have just removed that facility from their website, and have said that Microsoft Support will take people through a manual update process if necessary)
Lots of optional apps and services do require an internet connection, but these are optional parts of the experience. Some examples: