For the larger version, I would not cut the outer squares off to triangles. They could be trimmed to 1" beyond the edge of the quilt, the excess pressed in half, then folded to the back like a single-fold binding to finish the edge. If you cut them larger (sort of like oversizing a setting triangle), then the corners will overlap, and you can seam them together to get a continuous bias binding.
For this little one, you could use bias tape to finish it. It will give a finished edge with less bulk than a double-fold binging. A satin stitch would also work.
What to you plan to do with this sample? If it was mine, I would look at getting a picture frame and displaying it that way - I think it's gorgeous! You could use a dark frame with a white or cream mat; this would give the white border look that a couple people have mentioned. You might have to put some batting behind the mat to get it to sit correctly, and I probably wouldn't use glass in front of it.