Boxes Around My Download Vector Files

Question

When I originally downloaded this icon set, they opened normally, weren't legacy. However, now when I open the original icon files they are composed of boxes.
Let me try and explain- normally, you creat an amorphous vector shape with the pen tool and fill it in with a gradient right?
Now it is showing up as if each object is a square of gradient with a clipping mask around it- except it isn't actually a clipping mask and the actual boundaries of each object are square rather than the visible shape.
If I select the square using the white cursor, it deletes the fill. On a small amount of them, it will allow the option to remove a clipping mask- if I select "remove clipping mask" it becomes one big mess of squares. Most of them don't even have this option.
This is a big issue because I often deeply edit these files and am hindered by these boxes. Also, when I transfer them into InDesign, the transparancy is lost and the transparent parts of the boxes are turned white. A big issue!
Icons that I previously used and put in different ai files that I had created are still functioning normally, don't show the boxes, but seem to also have the problem of white instead of transparent in InDesign
None of the files I create have this issue, only these downloaded ones.
I have reset all my illustrator preferences and it didn't change anything.
Here is an example. One is an icon I had saved on a previous project, the other is straight out of the original file:
How do I get rid of these boxes?? What are they??

Answer 1

I didn't save them in any sort of format, this is how they came to me and, without editting the files, have begun to react differently when opened.
I think that does mean there's no way to "fix" a file that's been "squared"
Ok, so second part of the question, anyone know why all of my vector icons (even the non-expanded ones) are creating white boxes around themselves in InDesign? I have tried placing them and pasting them in every way possible. I'd just delete the white, but that destroys the shadows and effects as well.