Wayne Couzens, 48, was arrested on Tuesday after the 33-year-old disappeared when walking home to Brixton from Clapham in south London last week.
Ms Everard’s body was found in an area of woodland in Ashford, Kent, Westminster Magistrates Court heard.
Mr Couzens wore a grey tracksuit as he appeared for the brief hearing.
He appeared to have a red wound on his head and spoke quietly when asked to confirm his name, date of birth and address.
Chief Magistrate Paul Goldspring remanded him in custody to appear again at the Old Bailey on 16 March.
The court heard the officer, of Deal, Kent, is accused of kidnapping Ms Everard in the Poynders Road area of Clapham.
Prosecutor Zoe Martin said Ms Everard was reported missing by her boyfriend on 4 March.
She had visited a friend in the Clapham Junction area on 3 March and left at around 21:00 to make what was roughly a 2.5 mile (4 km) journey home.
She called her boyfriend for around 14 minutes, and was seen alone on CCTV at 21:15 GMT and again on a camera at 21:38, Ms Martin said.
Her body was found on Wednesday inside a builder’s bag and later identified through the use of dental records, the hearing was told.
A post-mortem examination has taken place but no cause of death was given in court.
Mr Couzens joined the Metropolitan Police in 2018, most recently serving in the Parliamentary and Diplomatic Protection Command, an armed unit responsible for guarding the Parliamentary estate and embassies in London.
His main job was uniformed patrol of diplomatic buildings and Scotland Yard said he was not on duty at the time of Ms Everard’s disappearance.
He was taken to hospital for a second time in 48 hours on Friday for treatment to another head injury suffered in custody, before he was discharged and returned to a police station.
Mr Couzens was previously treated in hospital for a separate head wound on Thursday, also sustained in custody when he was alone in his cell.
Meanwhile, police in dry suits were seen continuing to search around the area where Ms Everard’s body was found near Ashford on Saturday.