How to update Samsung SSD firmware?

This seems like such a simple thing to do, but I'm getting stuck. My computer suddenly stopped talking to my SSD a few weeks ago, and after some help on here I was able to determine it was due to a bad internal SATA cable. But unfortunately, there still appears to be an issue with the SSD: it will eject and remount repeatedly when I connect it in its USB enclosure. I realized the firmware was out of date, and I am hoping that will fix the issue.

But I'm not sure exactly what steps need to be taken. I downloaded the update, an ISO file, and I have burned it do a disc, but the disc won't start the install process. I've read I need to boot from this disc (hold Alt when powering up), but doing this just takes me to recovery mode and I don't know what to do from there. It seems like this whole process would be a lot easier on Windows. Could I just install the update through a Windows computer even though the drive is formatted for my Mac?

Thanks for your help!