Hello Christian,
You may not be able to achieve a full square image. It just that background images depends on its container element. I would like you to try this demo here: https://davidwalsh.name/demo/background-size.html which would explain how background image works.
We use the background-image: cover;
which would cover the whole area of the container. This is responsive enough which will always cover the whole area. The only thing to consider here is that if your image size is smaller, it will be stretch out and if you have a bigger image, it will get cut off.
Yes, we can use background-size: 100% 100%;
which is also responsive but your image will either be stretch out or get squeezed depending on the size of the container.
Yes, we can also use background-size: 100% auto;
but then as soon as you resize your browser or on smaller screen sizes, you will probably have white spaces around the image.
Or we can also use background-size: contain;
but then this is not a good choice because you will see white spaces around your image.
To better assist what you are trying to do, would you mind sending us the URL of the page that you are working on? Maybe we can give you some ideas on how you can have your background image as responsive as it could possibly be.
Thank you for your understanding.