Is 0-1 integer linear programming NP-hard when $c^T$ is the all-ones vector?
We can encode satisfiability of a SAT instance as the feasibility of a 0-1 integer linear program. For feasibility, the objective function doesn't matter, so you can impose whatever constraint you wish on it.

For an example of how to express boolean or, boolean and, and boolean negation in a 0-1 integer linear program, see Express boolean logic operations in zero-one integer linear programming (ILP).  This is all that's needed to express a SAT instance as a 0-1 integer linear program.