Scott Adams (creator of Dilbert) on his popular blog stated this in a post about confirmation bias:
Case in point, just yesterday Trump suggested that perhaps flag burners should be punished. That’s clearly dictator talk! (Except that Hillary Clinton proposed that actual law in 2005)
Did Hillary Clinton propose a law in 2005 to punish flag burners?
The proposed law co-sponsored by Clinton in 2005 prohibits flag burning under specific circumstances. It does not attempt to make flag burning illegal in general (i.e. outside of those specific circumstances). Clinton has voted against a proposed amendment to the Constitution that would allow Congress to ban flag burning in general.
In 2005, Clinton co-sponsored the Flag Protection Act of 2005, whose summary is as follows:
Flag Protection Act of 2005 - Amends the federal criminal code to revise provisions regarding desecration of the flag to prohibit: (1) destroying or damaging a U.S. flag with the primary purpose and intent to incite or produce imminent violence or a breach of the peace; (2) intentionally threatening or intimidating any person, or group of persons, by burning a U.S. flag; or (3) stealing or knowingly converting the use of a U.S. flag belonging to the United States, or belonging to another person on U.S. lands, and intentionally destroying or damaging that flag.
It seems like a reaction to ongoing efforts to pass a constitutional amendment allowing Congress to ban flag burning, a move that the co-sponsors of the 2005 act oppose and voted against. Its text notes:
the Bill of Rights is a guarantee of those freedoms and should not be amended in a manner that could be interpreted to restrict freedom, a course that is regularly resorted to by authoritarian governments which fear freedom and not by free and democratic nations
but suggests that flag burning with intent to incite violence is not protected by the Constitution (unlike flag burning as political expression, which is protected):
destruction of the flag of the United States can be intended to incite a violent response rather than make a political statement and such conduct is outside the protections afforded by the first amendment to the Constitution.
Note that the distinction between
is an important one. The Flag Protection Act proposed in 2005 was essentially a replacement for a previous Flag Protection Act that banned flag burning in general:
Whoever knowingly mutilates, defaces, physically defiles, burns, maintains on the floor or ground, or tramples upon any flag of the United States shall be fined under this title or imprisoned for not more than one year, or both.
but this was deemed unconstitutional in United States v. Eichman, 496 U.S. 310 in 1990, on the basis of their earlier (1989) decision in Texas v. Johnson, 491 U.S. 397. There, the Supreme Court noted that
an important governmental interest in regulating nonspeech can justify incidental limitations on First Amendment freedoms when speech and nonspeech elements are combined in the same course of conduct
but that in the case under consideration, this did not apply:
An interest in preventing breaches of the peace is not implicated on this record.
and further noted that
Expression may not be prohibited [p398] on the basis that an audience that takes serious offense to the expression may disturb the peace, since the Government cannot assume that every expression of a provocative idea will incite a riot, but must look to the actual circumstances surrounding the expression.
Clinton was not one of the 59 co-sponsors of a proposed amendment to the constitution that would allow Congress to ban flag burning in general:
Proposing an amendment to the Constitution of the United States authorizing Congress to prohibit the physical desecration of the flag of the United States.
and she also voted against it. The proposed amendment failed to pass by one vote.
