If my boyfriend moves in does my husband have grounds to prove me an unfit mother?

First, with regard to the financial issues you are having, you may be able to file for divorce with an affidavit of indigency and have the divorce filing fee waived.  

With regard to having your boyfriend move in, I would not advise doing so prior to at least filing for divorce.  While having someone move in would not automatically trigger the court finding you unfit, it is a point your husband could raise in a custody battle.  Many courts do not like seeing children exposed to other individuals during the divorce since it can create confusion for the children.  

Also, in Massachusetts, while it is a no fault state (you do not need a reason to get divorced) your husband may counterclaim on the basis of adultery when you do file for divorce.  Having a live in parter would be strong evidence that adultery is occuring.