It was the second such death this month.
The agency said it has notified the Department of Homeland Security inspector general about Felipe's death.
He also wanted a new bicycle, like another boy in the village purchased with money sent after his father went to work in the United States.
Many of them say their goal is to settle in the U.S. despite warnings by U.S. officials that anyone found entering the country illegally will face arrest, prosecution and deportation.
The death of the seven-year-old girl fueled criticism of President Donald Trump's immigration policies.
Thousands of migrants continue to flood the U.S. southern border each month - and trapped in the mix are children. He is the second migrant child to die in the care of the U.S. authorities this month.
An eight-year-old migrant boy has died in USA government custody.
She was said to have had a body temperature of 105.7F (40.9C).
Both Felipe and Jakelin came from rural communities with extreme poverty. "Our thoughts and prayers are with this family during this very hard time". He died around midnight.
But now, as more children cross the border, "We're doing dozens of hospital trips every single day with children that have fevers or manifest other medical conditions", McAleenan said Wednesday. "This is why I asked Congress on Tuesday to change our laws so that the United States is not incentivizing families to take this unsafe path", McAleenan said.
But McAleenan's pleas for Congress to assist border patrol in the care and protection of migrants comes during a partial government shutdown.
Felipe "always wanted a bicycle", Lucas said, and the family thought he would be better off in the United States.
Gomez was detained with his 47-year-old father at a crossing in El Paso, Texas, on December 18 and had been transferred to a New Mexico medical center showing signs of sickness on Monday, the CBP said.
At 10 p.m., Homeland Security officials said, Felipe appeared lethargic and nauseated, so the agents took him back to the hospital.
An autopsy performed on 8-year-old Felipe shows he had the flu, the New Mexico Office of the Medical Investigator said in a statement late Thursday, adding that more tests need to be done before a cause of death can be determined.
The Washington Post notes that the timeline by CBP has changed multiple times, regarding the events surrounding Alonzo-Gomez's death, first saying that he died on Christmas day, then clarifying that he died on Christmas Eve.
Agustin Gomez Perez was 47 and in debt, and that path would only deepen his obligations. It lacked basic services.
The federal agency is under attack after Felipe Gómez Alonzo, a young Guatemalan boy, died on Monday evening while being detained at a center in New Mexico.
The altar stood to one side of a room with cement walls that serves as a bedroom and living area for Alonzo and her three surviving children.