No, it is not correct because the main verb after a modal verb should be in the present simple tense (i.e. will lay). Next, you are willing to give up your life for someone. To express this meaning, the phrasal verb is "lay down".
Therefore, the sentence is: I will lay down my life for him.
On a separate note, "lay" (which means to deposit something on a surface) is in the present simple tense; "laid" is in the past simple tense. This should not be confused with another verb "lay" (which is the past simple tense of "lie" which means to have one's entire body on a surface. Example: He lies on the bed. He lay on his bed last night.)
I hope this helps.