If your boyfriend is not the father of your son, you can claim your son if your son does not provide more than half of his own support and is under 19 years of age or under 24 years of age and a full time student.  If both of you are his parents you can decide who will claim him.

As for the daughter who is a child for both you and your boyfriend, you can decide between you who will claim her.

Review the information in this TurboTax article for more information by clicking on the hyperlink. Rules for Claiming a Dependent

The IRS will make the decision under the Tie-Breaker Rules only if you both claim a child (noted and linked in the article).