No. You can not, usually, claim the EIC, if the child is not your dependent.
You may have heard of situations where that is allowed. They do not appear to be applicable in your case. If TurboTax is saying you qualify for the EIC, you may have indicated, in the interview, that the other parent is claiming the child, not that the child is claiming himself. See below for an explanation*. Another possibility is that you have indicated that the child is self supporting. See below for a discussion on that**.
But, more importantly, as others have already pointed out, your son does qualify as your dependent (he's under 19, lived with you more than 6 months, and did not provide more than half his own support).
Equally important; with the tax law change, effective 2018, most dependents will get the same refund whether they claim themselves or not. The personal exemption has been eliminated and the standard deduction increased.
And he will get the $1200 stimulus check, even if he is your dependent (as long as he's not your dependent for 2020). He'll just have to wait until early 2021. See below for an explanation***.
You should file a tax return, claiming him and the EIC. You cannot e-file. You will have to mail in your return. If he will not voluntarily amend, the IRS will eventually make him amend (and make you prove your claim). But amending will not (most likely) change his refund.
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* There is a special rule in the case of divorced & separated (including never married) parents. When the non-custodial parent is claiming the child as a dependent/exemption/child tax credit; the custodial parent is still allowed to claim the same child for Earned Income Credit, Head of Household filing status, and day care credit. This "splitting of the child" is not available to parents who lived together at any time during the last 6 months of the year; then only one of you can claim the child for any tax reasons. The tax benefits may not be split in any other manner.
Note in particular that the non-custodial parent can never claim the Earned Income Credit, Head of Household filing status or the day care credit, based on that child , even when the custodial parent has released the exemption to him.
** There is a special rule that doesn't come up very often. A parent qualifies for the EIC, even if a child, that lives with him, is self supporting. You say "The dad prepared tax return in turbotax, and he choose did not support the boy more than 6 months". I believe the exact wording of the question is "Did (Child's name) pay for more than half his living expenses". If you answer yes to that question, and also indicate that the child is under 19, and lived with you more than half the year; then TT is correctly giving you the EIC, even though he does not qualify as your dependent. So, your friend is correct, it could meet all tests.
***Under the CARES Act, if you are claimed as a dependent on someone else’s return you cannot receive a stimulus check, in 2020. If you were claimed as a dependent for 2019, but will not be for 2020, you will most likely get it in 2021..
"In essence, the stimulus check acts as an advance of your 2020 income tax refund. This means when you prepare your 2020 income tax return, there will be a line to include the section 6428 credit. The credit on your 2020 return is subtracted by any amount received as a stimulus check in 2020. If the amount you received as a stimulus check is less than the credit you are due, the difference will be included as part of your 2020 refund. If you have been overpaid by receiving the stimulus check, however, you will not be required to return any excess amount".
