My workstation's a Ubuntu Gnome desktop, hence the GTKs' and apt commands below. Adapt it to your need knowing WMWare stuff will work no matter your (mainstream) Linux OS.
Download the ISO from the official website. Product's name is VMware vSphere Hypervisor, file is described as ESXi ISO image.Here's where I found it in VMWare's website maze : https://my.vmware.com/group/vmware/evalcenter?p=free-esxi6. I had a license, maybe that can be downloaded for a free trial.
Create a live USB stick with it. I used
gnome-multi-writer because you can't have a simpler GUI that just works... and Ubuntu's default software for Live USB,
usb-creator-gtk didn't worked at the time.
It's the simplest way I found for some backup before upgrading. Download VMWare vSphere CLI sources from this link, with the same version as your ESXi : https://code.vmware.com/web/dp/tool/vsphere-cli. Then install it.
sudo apt install -y perl-doc libperl-uuid cd Downloads tar xf VMware-vSphere-CLI-5.5.*.x86_64.tar.gz cd vmware-vsphere-cli-distrib sudo ./vmware-install.pl
vicfg-cfgbackup --server=10.25.77.10 --username=root -s esxi10.firmware.backup
-q for a no prompt command.
To restore, use the exact same command with
Warn your users, shutdown your VMs, switch your ESXi to maintenance mode just to be sure.
Plug your USB stick with ESXi ISO on your physical server.
Shutdown server, start it, go in the (uefi) bios, change boot order to start on USB stick. Reboot again.
You should now boot on a ESXI installer. Note it's almost the same look and feel than the usual ESXi boot screen, only the title is different.
Then go for the install, select the hard drive were ESXi 5.5 was installed (F1 key give you more details about a disk, including whether ESXi is installed here or not).
Once the upgrade start, which will take a couple of minutes, then reboot, then you should be done.