First of all, they must have a social security number or individual taxpayer identification number to be claimed on a tax return. In addition, they need to meet the following requirements:
1. The person cannot be your qualifying child or the qualifying child of any other taxpayer. A child is not the qualifying child of any other taxpayer if the child's parent (or any other person for whom the child is defined as a qualifying child) is not required to file an income tax return or files an income tax return only to get a refund on income tax withheld.
2. The person either (a) must be related to you or (b) must live with you all year as a member of your household.
3. The person's gross taxable income for the year must be less than $4,050 in 2016.
4. You must provide more than half of the person's total support for the year.
5. The person must be a U.S. citizen, U.S. resident alien, U.S. national, or a resident of Canada or Mexico.
Generally you are not required to claim someone on a tax return even if the could be claimed. I don't know what bearing, if any, your sponsorship for a green card application would have.
