The High Court's order came in a suo moto rule on Thursday after a lawyer drew the court's attention to a report run by a national daily over the incident in Bogra's Sherpur Upazila.
The court has summoned Sherpur's Upazila Family Planning Officer Mokhlesur Rahman and the hospital's Senior Staff Nurse Shushma Rani.
The two have to appear before the court on Dec 14 and explain themselves.
Bogra's deputy commissioner has been ordered to probe the incident and file his findings the same day.
The HC bench of justices Quazi Reza-Ul Hoque and Mohammad Ullah also issued a rule asking why legal actions would not be taken against those responsible.
The health secretary, Bogra deputy commissioner, civil surgeon, superintendent of police, Sherpur Upazila administration chief (UNO) and the hospital's resident doctor have been told to come up with explanations.
On Thursday, a Bengali daily ran a report saying Mazeda Begum, who was on labour, was taken to the Sherpur Upazila Health Complex around 11pm. She was admitted to the hospital.
But Senior Staff Nurse Shushma Rani advised them to go to the nearby Modern Clinic and went to sleep, says the report.
Around 12:45am, the hospital's staffer Parvin Bibi forced them to leave, according to the news report.
Begum gave birth to a child near the hospital's entrance under the open sky. A night guard and a passer by then went inside the hospital to get the staff.
Resident physician Pial and Nurse Shushma later declared the new born dead.