hi Miner38
imho it is not a good idea to add firefox to a tinycore .iso file via ezremaster.
One of the first extensions I download is  firefox_getLatest.tcz - and in the Terminal run  
tc@box:~$ firefox_getLatest.sh
When this is done, you are prompted to add "firefox.tcz" to your onboot.lst 
tc@box:~$ echo "firefox.tcz" >> /ext/sysconfig/tcedir/onboot.lst
and reboot. Up from now your firefox  can be started by its wBar icon.
In your desktop menu under  applications two sub-menus are added:
firefox  
  to start yor browser 
firefox_getLatest  to update
  to download and install the newest firefox version. Use  firefox_getLatest from Desktop Menu every time Mozilla announces a new version,   but do not use the link that Mozilla proposes.
This is independent from the main updating tool for tinycore and should be done before:
tc@box:~$ tce-update
This updates all of the tinycore extensions it finds to be updateable, but it complains in red letters to be unable to update firefox. You can ignore this warning.
The other browsers like  chromium, brave, firefox-esr, or the old ones are covered by  tce-update .
Have fun