We are not married and are new parents (to twin girls!) so we weren't certain what to do. I've always filed and taken care of my taxes and he unfortunately hasn't... I did withdraw a 401k of about $3300 and I only made just over $10,000 from employment income. Am I correct in thinking that I am the one who should claim them? Especially because he didn't file last year?
That depends. You're both entitled to claim both the children or either of them.
If the twins are the first children in the household, it presents new opportunities for you in terms of refund.
One is the Head of Household filing status. The other is Earned Income Credit and Child Tax Credit for your children.
Without knowing the whole situation from either of you, you would do well if you let TurboTax tell you what the refund would be with you claiming both children, your boyfriend claiming both or you each claiming one.
Remember that there can only be one Head of Household per household, so check that out also.
Finally, that fact that your boyfriend didn't file last year is not an issue if he wasn't required to file. Many times people who don't file are due a refund.
Here are a couple of links that may help you:
Do I qualify for Head of Household? - TurboTax Support