I think the squared image is more a choice for simplicity.
There are two types of convolutional neural networks
- Traditional CNNs: CNNs that have fully connected layers at the end, and
- fully convolutional networks (FCNs): they are only made of convolutional layers (and subsampling and upsampling layers), so they do not contain fully connected layers
With traditional CNNs, the inputs always need to have the same shape, because you flatten the last convolutional layer, with a fixed size. As the flatten layer has a fixed size, the feature map shape from the layer before has to be the same shape, and so the inputs (images) also have to.
However, in FCN, you don't flatten the last convolutional layer, so you don't need a fixed feature map shape, and so you don't need an input with a fixed size.
In both cases, you don't need a squared image. You just have to be careful in the case you use CNN with a fully connected layer, to have the right shape for the flatten layer.
For instance, if you have an input size $320 \times 160$, and you have 3 pooling layers, so your output in the last convolutional layer is $40 \times 20 \times c$ (with $c$ the number of filters/channels), then you just need the flatten layer to have $40*20*c$ neurons.
If you create a new network, just design it to handle a rectangle image.
If you want to use an already pre-trained one, I think the better choice is to resize the image.
If the information in the cropped parts is important, maybe your prediction can be wrong (it depends if the object of interest is in the parts of the image that is cropped). Actually, in Yolo (an object recognition network), images are resized if they don't fit the input requirements. See figure 1 of the YOLO paper. It's because you don't need a high resolution to detect an object (for example, the CIFAR dataset has images of shape $32 \times 32$, but the network can still predict the correct label). So, I think that resizing your image may not affect the prediction much (unless the new size is very different from the original)