For what it's worth, at this moment the documentation for MUCXpresso IDE on:
Host operating systems:
which (indirectly) suggests that the IDE won't run under macOS 11 (Big Sur). Perhaps the document could be updated...
It does install and run, but in use it is pretty poor.
Half the time, the screen doesn't update properly, so your editor doesn't show the actual code you are editing. Same is true for most other screens, you get distortions and incorrect values until a redraw of the screen is forced (usually by scrolling the window).
I wish I could downgrade back to Catalina and wait for MCUXpresso to use the latest Eclipse which has fixes for Big Sur.
It is on an Intel Mac (iMac 5k). I have been living with it, but I have made a few errors specifically because of the UI bugs, such as editing the wrong line of code because the UI hadn't refreshed.
It may also be my imagination, but my build times seem slower on Big Sur than they did on Catalina.
I've noticed the screen update issues, but as I use an external editor 95% of the time, the impact is minimal. (In fact, I thought it was just a quirk of my ancient MacBook Pro.) But I had no difficulty installing or launching it.
And FWIW, it's not just MCUXpresso -- I've seen the lack of screen update on other Eclipse-based IDEs.
Assuming you mean on a Mac with Intel silicon, then yes it does run, although installation is a pain as the package is not notarized (but it is signed). It is possible that it may run on a Mac with Apple Silicon using Rosetta2, but I've not seen anybody say they have even tried it.
This FAQ explains how to install on Catalina, but it is similar for Big Sur.