Is it permissible for a Muslim residing in America to benefit from free church services, such as food, if he is of limited income?

There is no harm in accepting assistance from the state or the church, as long as it does not lead to compromising one's religion, affirming a wrong, or negatively influencing one's children. However, abstaining from it is better, because "the upper hand is better than the lower hand." Seeking charity is blameworthy except for the one in dire need. So, whoever is not in urgent need of assistance should refrain from seeking it.