Not a lawyer, but I was trained in First Amendment law as part of my degree.
The First Amendment does not apply to individuals or businesses limiting other individuals or businesses. First Amendment doesn't enter into this discussion. Contracts aren't speech. They're agreements between people, and the government does limit what types of things you can compel another person to do in a contract. Frequently, Congress has acted to limit asymetries of power... notice that this bill affects FORM CONTRACTS only ... that means the ones that are created for mass signatures... if you want to have a personalized contract -- one where you sign and then a company rep personally countersigns -- then I think this doesn't apply. But for that last bit, you would need to check with a lawyer.