If your not married, can you claim a child as your stepchild?

Your fiance can claim you and your children as his dependents (as qualifying relatives) if each of you pass all the following tests:

1. The person can't be his qualifying child or the qualifying child of any other taxpayer. 
2. The person must live with him all year as a member of his household (and the relationship must not violate local law). 
3. The person's gross income for the year must be less than $4,050. 
4. He must provide more than half of the person's total support for the year.
5. The person is a U.S. citizen, U.S. resident alien, U.S. national.
6. The person is not a married person who files a joint return unless that joint return is filed only to claim a refund of withheld income tax or estimated tax paid.

Claiming your children as dependents does not qualify him to file as Head of Household as they are not related to him. He must file Single.

I do not think its fair at all. We support these kids all year and because your not married or have custody of these kids you get nothing. All thes states just want your money. So take peoples money. 

As MinhT mentions he can claim all dependents as long as the tests are met. His filing status is just not Head of Household.  

So i can get the Earned income credit for the children or that is only if i am legally married to my fiancee because my tax preparer is telling me i cant get the earned income credit just put on as dependents by the way I live in New York State

You cannot claim your significant other's children if you are not married unless you have already legally adopted them, even in New York state.  

@MandE18