I live with my boyfriend and we have 2 children together (we are not married). I understand that we cannot both claim head of household, but I make less money, so it's more beneficial for me to claim our children as dependents as I get earned income credit. But since he makes more money, can he claim head of household with me claiming both kids?
In your case you are one household. The situation described by Heather applies to 2 unrelated families. To file as HOH you must have a qualifying child so you cannot claim both children if he wants to file as HOH.
One of the requirements for Head of Household (HOH) filing status state that you must have paid more than half the cost to run your home for that year. Although, there are exceptions to this rule. If there are two households sharing one house, and each household can prove that they exist as separate households, and meet the other requirements to file as Head of Household, then two taxpayers living at the same address could qualify for HOH filing status. For example, if two unrelated taxpayers each with children occupy living quarters in common and each taxpayer pays more than one-half of the household costs incurred by each respective family, each taxpayer will be treated as maintaining a separate household.
Please see the following IRS link for more information: https://www.irs.gov/pub/irs-wd/1998-041.pdf
Please see the following TurboTax FAQ for more information on what the qualifications are for the filing status of Head of Household: