"In a criminal case, can the government make you testify if you claim your Fifth Amendment privilege against self-incrimination?"
From 1870 until 1970, in order to compel a person to testify who had claimed the Fifth Amendment, the government had to grant what was called "transactional" immunity.
So, if you were granted immunity you could be made to testify but there was no way that the government prosecutors could ever charge you with any crime related to your testimony. This the United States Supreme Court said was the only constitutional way to grant you immunity and to make you talk.
In 1970, however, the United States Supreme Court reversed its position and declared that "transactional" immunity was no longer necessary. A more limited form of immunity known as "use and derivative use" immunity could be used instead.
The governments granted immunity meant that you could be made to testify despite your claim of the Fifth Amendment, and the government could still charge you with a crime if it did show that the material it was using against you did not come from your testimony but from some outside source. As Part of the Omnibus crime control act of 1970 this new immunity became widespread and was used not only against organized crime, but for example, the later Attorney General of the United States, John Mitchell.
For questions regarding your criminal case, contact a knowledgeable Illinois lawyer today.