Both are arguably correct but me is preferable. It can only be ‘my boyfriend and I’ if one is trying to say ‘this is my boyfriend and I’, accepting that the verb ‘to be’ takes the nominative case. But I presume you don’t usually say ‘this is I’ because that sounds like an affectation.

You would never say this is a picture of I. So you mustn’t say ‘this is a picture of my boyfriend and I.’