Maquin and her father were taken to the Antelope Wells port of entry after turning themselves in to CBP officers in early December.
The CBP chief called the young boy's death "a tragic loss".
DHS wouldn't say how many children are in Border Patrol custody.
Customs and Border Protection said the child's official cause of death has not been determined.
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"This is inexcusable", she said in a statement Tuesday.
The boy had been in CBP custody on Christmas Eve when a United States border patrol agent noticed he appeared unwell, and was transferred to a medical centre where he was diagnosed with a cold and fever, according border officials.
The children's deaths have highlighted the life-and-death challenges that migrants face in attempting to come to the United States and caused CBP to announce several policy changes around its care of children under 10 years old. Both were taken to a medical centre in New Mexico's Alamogordo, where the child was diagnosed with a common cold and fever and released.
She became sick on the bus ride to a border patrol station and arrived with a 41-degree Celsius fever. She was twice revived by Border Patrol emergency medical technicians or EMTs and airlifted to a Texas hospital.
Felipe was apprehended with his father at a location about 3 miles west of the Paso Del Norte port of entry in El Paso, Texas for illegal entry.
CBP released a timeline of the days before the boy's death.
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A 7-year-old Guatemalan girl died of dehydration and septic shock while in government custody earlier this month.
A CBP spokesperson could not immediately answer how many children are in the agency's custody. He died at 11:48 p.m. Monday, the agency said.
On Sunday, because of overcrowding at the El Paso facility they were in, they were shipped to a facility in New Mexico and on Monday - six days after they were arrested - CBP contacted U.S. Immigration and Customs Enforcement for a more permanent placement. The child was observed for an additional 90 minutes in the hospital and was given prescriptions for amoxicillin and Ibuprofen, before he was released from the hospital in the afternoon of December 24.
Guatemala's Foreign Ministry said its consul in Phoenix was seeking to interview the boy's father, to whom it pledged to give all necessary consular assistance and protection.
A congressional oversight trip last week to the CBP station where the girl and her father were detained raised "serious concerns" about facility conditions and the "lack of adequate medical supplies, equipment and resources to properly treat migrants and the agents working there", Castro said.
Rep. -elect Xochitl Torres Small, who will represent El Paso in the House of Representatives, called for a thorough investigation into the children's deaths and medical treatment along the border.
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"The cause of death is under investigation and medical reports have been requested to clarify the cause of death of the child", the ministry added.