Boehner and his GOP House leadership said they would proceed with votes, including a rare Saturday session, on piecemeal spending legislation to fund popular programs while the shutdown continues.
Democrats oppose the incremental approach, saying it amounts to conservatives pushing it to fund programs and services they like.
House Republican leaders sought to depict the legislative wrangling as Democrats harming federal workers and denying services such as national parks and veterans affairs by voting against the limited spending proposals focused on them.
Other piecemeal spending measures will provide funding for the Federal Emergency Management Agency and the National Weather Service with a tropical storm brewing in the Gulf of Mexico, as well as the Head Start program, Majority Leader Eric Cantor of Virginia said Friday.
Obama would veto such measures if they reached his desk, the White House has announced.