Your host isn't bound by the first amendment unless it's a US government entity (which I find extremely unlikely). The US First Amendment restricts
government restrictions on speech, not private entities.
What you can do is register a DMCA claims agent (see http://www.copyright.gov/onlinesp/ ). The DMCA provides a "safe harbor" for web sites. If you have a registered DMCA agent, there's a set process for the agent (which may well be you from what I recall) to receive a takedown request (content, text, images) and a time period to respond. You in turn will need to take some sort of action.
If you don't have a DMCA registered agent you don't necessarily lose "safe harbor" but it's more difficult to block a service provider from just defaulting to removing content and/or sites.