No. Your Stimulus is not based on filing with a SSN or ITIN.
Your employer does not determine who should get the Stimulus payment. It is a government stipend with clear qualification requirements. If you have an adjusted gross income (AGI) of up to $75,000 ($150,000 married filing jointly), you should be eligible for the full amount of the recovery rebate.
CA EITC is not related to the Stimulus requirement. You said your Form 1099NEC does not qualify for CAEITC and that is not correct. To qualify, you must have income from employment, self-employment, or employer-paid disability benefits received prior to retirement. There is no limit to the number of times you can claim an EITC; you can claim one every year that you qualify.
Perhaps you mean Form 1099G Unemployment Compensation does not qualify for EITC because that is unearned income and does not qualify for the credit.
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