SquareIt takes your full-size image and turns it into a square without trimming the edges or resizing, so you don't have to cut anything. To fill in the extra space, there are multiple background colors to choose from, and you can apply one of several filters, as well, to complete the perfect look.
This is a no-frills app that opens directly to the working screen. Through the control bar at the bottom, you can access your photo library, or you can take a new photo without leaving the app. Then, tap the palette icon to choose your background color from a selection of 30. Depending on whether you've taken a horizontal or vertical photo, the color you choose will appear as bars along the top and bottom or along the sides of your image. Next, select the filter you'd like from a list of 13 by tapping the Heart icon. When you go to pick the filter, the background color will disappear, but it will come back once you've applied the filter and returned to the main screen. This is a bit of a drawback, though, since you can't see what the filter and the background look like together until you apply the filter.
SquareIt works quickly, and its streamlined, efficient interface makes the lack of a tutorial or Help file really not much of a problem. It's easy enough to find all of the features after a minute or two of playing around with this app, and it's free.