This can currently be taken as a rumor at best, but a new trademark was filed by Square Enix recently which only said the name “Square” on the file. This was found by Destructoid initially and scanned the document which you can see below. Square Enix may not be going through a name change but this evidence is a bit strange.
Over the course of the companies history it was titled on games as either Square or SquareSoft but when they purchased Enix they changed their name to Square Enix and it has been that way ever since. We have reached out to our contacts at Square Enix in the hopes for some clarification on this issue.