You are only eligible to receive the Earned Income Tax Credit if you have taxable income from work. The Social Security benefits you received are not income from work.
If the Social Security benefits is the Only income to be reported on the tax return and there is no federal income taxes being withheld, then there is no reason or need to file a tax return..
Unless you also received a Form 1095-A for your health insurance through one of the State Marketplace Exchanges or from healthcare.gov then you must file a tax return.
