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.

While this may be excessive for your situation, you could try automation software to make your own login window, get the unique user credentials, then launch SSMS with the correct command line options. Obviously, you'll want to prevent users from running SSMS directly and you can do that through Windows group policy or through registry modifications. Or if you get a bit creative with your scripting, AutoHotKey could do that too.

1. Software to automate almost any Windows action - AutoHotKey
2. To see the command line options for SSMS, type 'ssms.exe /?'
3. To learn how to prevent users from running a program, google 'Windows prevent users from running a program'.

It will take a while to get through all that, but then you'll have more options for how to get unique things done quickly.

Ps. I don't condone software piracy or license violations at all. Use the above advice at your own peril. These are standard tools that any good Windows administrator should already know how to do or should have in their bag of tricks.

You can write a small tool that would search for the necessary window and send a message to it to disable the control. Or choose one of the existing tools.

Quick search found this question: https://stackoverflow.com/questions/11833500/preventing-cross-process-sendmessage-calls

with this link in it: https://www.raymond.cc/blog/how-to-enable-and-access-disabled-grayed-out-buttons-windows-and-checkboxes/

Of course, any such tool will be brittle. Maybe, instead, make a backup of file where this password is stored (SqlStudio.bin) and overwrite it with a fresh copy each time a user logs into Windows. You said:

But few of them accidentally click that checkbox, so each time we need to clear the SqlStudio.bin to solve the problem.

Rather than handling it as an exceptional situation, handle it as a normal situation and clear the SqlStudio.bin with every log on.

Alternatively, you can try to make that file read-only / deny write permissions to it.