No you cannot claim your GF's son.   In order to claim a child to whom you are not related you would have to live with the child the whole year.

I though it was half a year was the cut off,  and could dhhs take away there insurance if i do

No comment on the food stamps or medical care---we cannot answer questions about that.  But you cannot claim a child who is unrelated to you as your dependent  on your tax return unless the child lived with you for the ENTIRE year.  And you say you are there some of the time.  You cannot get stimulus money for the child if that is what you are thinking.  His mother can, though.  Even if you claimed the child all you would "get" is a $500 credit for other dependents to lower your own tax bill.   You cannot claim her child to file as Head of Household or to get earned income credit.  If his mother did not work she cannot get child tax credit or earned income credit, but she can get the stimulus money for herself and her child.

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@micalmeta wrote:

I though it was half a year was the cut off,  and could dhhs take away there insurance if i do

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The more than half the year is the requirement for claiming *your* child.   Claiming an unrelated child is the entire year.

---Tests to be a Qualifying Relative (& Unrelated Persons)---
(Must meet ALL of these tests to be a dependent)

1. The person cannot be your qualifying child or the qualifying child of any other taxpayer.

2. The person either must be related to you, or must live with you all year (all 365 days - There are exceptions for temporary absences such as school, illness, business, vacation, military service) as a member of your household.

3. The person's gross income for the year must be less than $4,300 (tax-exempt income, such as certain social security benefits, is not included in gross income)

4. You must provide more than half of the person's total support** for the year.

5. The person is not filing a joint return.

In any case, the person must be a U.S. citizen, U.S. resident alien, U.S. national, or a resident of Canada or Mexico

The above is simplified; see IRS Publication 501 for full information.
https://www.irs.gov/publications/p501#en_US_2018_publink1000220939

** Worksheet for determining support
https://www.irs.gov/publications/p501#en_US_2019_publink1000226268

As you can see ---there are two kinds of dependents---qualifying child and qualifying relative.

Your GF's son is not related to you so he cannot be your qualifying child.

In order to be your "qualifying relative" he would have to live with you the WHOLE year.

WHO CAN I CLAIM AS A DEPENDENT?

You can claim a child, relative, friend, fiance (etc.) as a dependent on your 2019 taxes as long as they meet the following requirements:

Qualifying child

• They are related to you.

• They cannot be claimed as a dependent by someone else.

• They are a U.S. citizen, resident alien, national, or a Canadian or   Mexican resident.

• They are not filing a joint return with their spouse.

• They are under the age of 19 (or 24 for full-time students).

• No age limit for permanently and totally disabled children

  •      They live with you for more than half the year (exceptions apply).

Qualifying relative

• They don't have to be related to you (despite the name).

• They cannot be claimed as a dependent by someone else.

• They are a U.S. citizen, resident alien, national, or a Canadian or Mexican resident.

• They are not filing a joint return with their spouse.

• They lived with you the entire year.

• They made less than $4300  (not counting Social Security)

• You provided more than half of their financial support. More info

When you add someone as a dependent, we'll ask a series of questions to make sure you can claim them.

Related Information:

• Does a dependent have to live with me?

• What does "financially support another person" mean?

• Can I claim a newborn baby?