The Quinnipiac poll shows that even most Republicans oppose a shutdown, by a 56-36 margin. It was likely numbers such as these that Boehner was referencing when he accused those who want to provoke a shutdown of being "unrealistic." 

In fact, a slim majority of voters don't even support cutting off funding to Planned Parenthood, regardless of whether it provokes a shutdown.

Fifty-two percent oppose cutting federal funds for the organization, which is prohibited from using public funds to provide abortions, while 41 percent support taking away the group's federal funding. 

The poll also found that 44 percent of Americans have a favorable view of Planned Parenthood, while 39 percent have an unfavorable view. 

"While voter support for Planned Parenthood is tepid, most American voters, especially women and Democrats, don't want to cut off federal funding," said Tim Malloy, assistant director of the Quinnipiac University Poll.

The group has long been targeted by anti-abortion forces who claim that laws that prohibit the federal government from funding abortions do not make a difference if the government is still financing groups that provide abortions.

Anti-abortion legislators argue that by financing other services that Planned Parenthood provides, such as contraceptives, breast and cervical cancer screenings and HPV tests, the government is allowing the group to devote its private resources to abortion. 

The Quinnipiac poll also gauged general American attitudes on abortion and found that the majority of Americans believe abortion should be legal in at least some instances.

While only 22 percent said it should always be legal, 35 percent said it should be legal in most cases. Another 28 said it should be illegal in most cases, while 12 percent said it should always be illegal.