I spent some time a few years back looking for precursors to "squircle". See https://www.johndcook.com/blog/2019/04/02/history-of-the-ter...

It appears that if you have four rods in a 2×2 formation, then the interior part was called a squircle in one paper.

There's also a "squircle" in Struble’s 1962 book “Nonlinear differential equations”, page 39.

See that link for more details.

While I didn't mention it, there was also a non-mathematical use of "squircle" as a magic trick to turn a square into a circle.