As J.A. said, this is not possible to do natively on the phone; all the stock video camera does is make those long videos.
However, there are a number of apps on the AppStore that do allow you to take square-shaped videos. I did a search at the AppStore for "square video" and got back 183 results. Some are to crop the video you shoot, while some allow you to directly shoot the video in a square frame. And, some are free (and some are not).
Hopefully, a search at the AppStore will help you decide which app will work for your purposes.