Is there a Fed. law saying you must take a lunch break within 6 hours of
clocking in?

No. Federal law does not require breaks of any kind regardless of the length of shift.

There are some states that do require lunch breaks after varying lengths of time, but New Mexico is not one of them.

I was fired for punching out 6 hours and 20 minuets into my shift. Was told
because I broke Federal law.

If they told you there was a Federal law that you must take a lunch break within 6 hours of the start of your shift, they were wrong.

However, if they want you to take a lunch within 6 hours of the start of your shift and you don't take it till 6 hours and 20 minutes in, and as a result they fire you, that is legal. (Petty, but legal.) There does not have to be a Federal law, or even a state law, broken before you can be fired.

Management was notified 30 minuets prior that I needed a relief person
as I was guarding an open Fire Exit and could not leave it. They sent my relief late.

There may (or may not) be a federal law that open fire exits must be guarded. If so however I am certain that law (should the law exist) does not actually mention the word "lunch". There is no right to lunch or breaks anywhere in federal law. It is perfectly legal under federal rules to require employees to work 24 hours shifts with no lunch. What little federal regulations exist that do talk about lunch and breaks do so in the context of how employees are paid.

The bottom line is that Federal law or no Federal law, based on the informatoin you have provided this was a legal termination.

Even though Management was the reason for being late to punch out. I could'nt even close and lock the door without them.

Yes. It falls under the unfair but not illegal category.