Just over 200 unionized public defenders at Neighborhood Defender Services across Harlem, Detroit and Texas said their management left them with no choice but to walk off the job Friday after contract negotiations over healthcare and pay remained at a standstill. Nearly all 120 Harlem-based public defender attorneys, social workers, paralegals and other legal professionals picketed in front of their 125th Street office from 10 a.m. to 3:15 p.m.…