How can I help convince my friend to think about professional counseling?

I’ve deeply hurt my boyfriend by breaking up with him. I still really care about his happiness and want to see him do well in the world. He is calling and texting me constantly, telling me he is so sad that he’s non-functional. I just can’t give him the support he needs because I’m dealing with an illness (bad flareup of a chronic condition) as well as sudden large-scale family commitments. When I tell him I can’t talk to him, he gets even sadder, telling me he can’t sleep, function or do anything but be sad. I really want to help him get the help I can’t give him. He says he is too embarrassed to ask for professional help, but his life is completely overtaken by this sadness and it has been for more than a month. Is there anything I can do? I want to be gentle with him but I am starting to fray a lot myself becuase I just can’t deal with the level of conact he’s trying to have with me.