I have seen the damage, first hand, that doing so can do to a child. The child will begin to look at the parent who’s being reprimanded as unimportant and see the one doing the reprimanding as having the back of the child. Not only does this create a role between the ill-reprimanding parent and child as allies but it creates a role for the other parent as being an opponent to both the child and the child’s ally.
In the absence of the child this should be addressed immediately. Explain that it is never okay to call you out in front of the child about the child’s care. Express that you are willing to listen to your boyfriend’s concerns with what you said, but that needs to be discussed away from the child. If your boyfriend plays any kind of a care-giver role with your child then you two should be discussing the “big stuff” before punishments or consequences are given, to make sure that you are on the same page. If you are not on the same page, you can get through all of the disagreeing without the child being present.
If in general, your boyfriend puts you down, degrades you or otherwise disrespects you, in front of the children, then you might want to reconsider the relationship in general.