What do I do when I was fired and job says I quit to still get unemployment

There is no requirement that you sign any document presented to you by your employer, especially if you do not agree with it.  If you sign this form, then you will probably be denied unemployment.  You can either ignore the letter, or reply back saying you were told you were terminated and asked to leave the job on Jan 2, and ask the employer why they want you to sign this letter under these circumstances.