Callaghan, who has been an assistant national for the past two years, will guide the US team through June and July as they defend their CONCACAF Gold Cup and Nations League titles.
As for Hudson, who is also an interim coach, he leaves the selection to take on a new, unspecified project.
The effective post remains vacant since the departure of Gregg Berhalter at the end of 2022.
"B.J. has been an integral part of the team for the past four years as this young team has grown and developed," said Matt Crocker, sporting director of the national team.
Callaghan, 41, is the oldest member of the US national team, having started in 2019 as a strategic analyst.
The United States will face Mexico in one of the semi-finals of the CONCACAF League of Nations, scheduled for June 15-18, in Las Vegas. The other game pits Canada against Panama.