Is it possible to disable the remember password checkbox in SSMS?
No, this is not possible, unless you reverse engineer SSMS and either hide the checkbox or make it a no-op, then recompile and deploy. And repeat every time you upgrade, apply a service pack, etc. (To be clear: this is not what I recommend.)

I have a better solution, though:

Why don't your 4-5 employees use different profiles? This takes a tiny bit more disk space, sure, but each profile gets its own SqlStudio.bin - then you could use Windows Authentication, too, which means they don't have to type their password or check a box, and it simplifies auditing, too, in case this isn't the most egregious thing they do.