If I'am living with a non-resident (we are not married) can we file our taxes jointly and claim child tax credit?

Negative.

To file Married Filing Jointly, you must be legally married or be considered to be married under a 'common law marriage' statute by the state where you live.

According to Wikipedia, only eleven states allow common law marriage; see link below:

Common-law marriage in the United States - Wikipedia, the free encyclopedia (http://en.wikipedia.org/wiki/Common-law_marriage_in_the_United_States)