No error messages or pop-up windows?
What specific mouse button issues?
Are you sure that the mouse/mouse button works on other computers? Eliminate the possibility that the mouse is broken.
Windows 10?
First try the built-in Windows troubleshooters:
"WIN" + "I" > Update & security > Troubleshoot.
You can also use
"WIN" + "I" > Devices to look into what may have gone astray.
Try removing the logitech gaming software via:
"WIN" + "I" > Apps > Apps & features
Then return to Logitech's website to verify the correct installer version and download again.
Reinstall per the applicable installation instructions. You may have had a buggy or corrupt version and/or missed some installation requirement.
Watch the windows, defaults, check boxes very carefully.