Not unless he feels he was terminated for his membership in a protected class, which a big thing you left out. In the OP, there is not mention of his membership in a protected class either. Without being able to draw that nexus, a trip/call to the EEOC is fruitless. Especially since 90% of cases that go to the EEOC are found not to be unlawful discrimination.

I know this because I am a federal EEO practitioner for the past 20 years and I currently serve as the EEO Director at a Federal agency.

Unless the original poster feels the termination was based on his status as a member of a protected class and has reason/ability to show that the given reason for termination is pretext for discrimination, he'd just be spinning his wheels at the EEOC.
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"To Each His Reach"

but.....

Fuck aliens.