Also wounded in the attack late on Thursday were 10 others including three police escorts and Esquivel's two sons, Mark Lawrence and Mark Anthony, police said.
So far, at least 20 people have been killed in connection with election campaigning.
"It could be a spur of the moment. These police officers are known to each other but it is unfortunate that they will be shooting at each other"
Ismael Rafanan, regional police chief
Chief Superintendent Ismael Rafanan, the regional police director, ordered the relief of the Jaen town police chief and deployment of more officers to prevent escalated violence.
He said the gunfight started when police acting as bodyguards for Rodolfo Antonino, the Nueva Ecija representative, opened fire on vehicles that failed to stop at a campaign rally roadblock.
Esquivel and Antonino are vying for the same post.
Elections in the Philippines are often marred by violence as rival groups use force to intimidate voters or political opponents.
"We're determining who among the policemen fired the first shots that triggered the exchange of fire," Rafanan said.
He said police protection for the politicians was authorised by the country's Commission on Elections.
"It could be a spur of the moment. These police officers are known to each other but it is unfortunate that they will be shooting at each other.
"They should act as the referees here," he added.
"Traditionally, the homestretch is the critical point when passions flare, old rivalries heat up and last-minute issues develop"
Oscar Calderon, national police chief
Nearly 87,000 candidates are vying for 17,000 national and local positions in the May 14 polls.
All 265 House of Representatives seats and half of the 24 Senate seats also are up for grabs.
"Keeping peace and order during the election period is the biggest challenge confronting us," Oscar Calderon, the national police director said.
"Traditionally, the homestretch is the critical point when passions flare, old rivalries heat up and last-minute issues develop," he said.
Police had already disarmed 52 out of 90 private armies and arrested 2,000 violators of a nationwide gun ban, Calderon added.
Next month, some 45 million voters will elect half the 24 seat Senate, 253 congressmen, and more than 18,000 local, district and provincial officials.