No, impeachment is for high crimes or misdeamenors. If not constitutionally eligible he would simply stop being president until such time as he is eligible. In the case of the natural born citizenship issue, that would be never.
It all really depends on 1) whether Speaker Boehner believes the claim that Obama lacks qualifications, and 2) whether Speaker Boehner is afraid that he won't feel bold enough to actually impeach if a committee were to prove that Obama lacks qualifications.