Can my boyfriend claim my kids he's taken care of the whole year?

Your boyfriend can only claim your children as dependents if they lived in his household for the entire year, except for temporary absences, AND he paid more that half their costs of living, AND you are not a taxpayer.

You are not a taxpayer if you don't file a tax return; or if you earned less than $12,400 and only file a tax return to claim a refund of withholding and claim no additional credits, deductions or dependents.  (If you are a taxpayer, he can't claim them even if you want to allow it.)

For the cost of living calculation, you must include all sources of support for the children's expenses -- including child support, support you provide, support provided by the state (section 8, SNAP, etc.) and support provided by other relatives.  Their costs of living include an equal share of rent and utilities, household food, plus their personal expenses for travel, entertainment, clothing and medical costs.  (For example, if your boyfriend pays 100% of the rent, food and utilities, equaling $1600 per month, then he gets to count $400 as support for each child, assuming 4 people total living in the home.)  There is a worksheet here to help figure out support costs. https://www.irs.gov/pub/irs-pdf/p501.pdf