My wife and I are both US citizens planning on giving birth in possibly France or Switzerland while I am there for work. How soon and what is required for us to be able to travel with the new baby within Europe? Does the baby need a US passport before being able to cross country borders even if it's from one one EU nation to another? I am assuming yes, especially if it's to or from Switzerland or the UK.
Switzerland is in the Schengen area, so there are no permanent immigration checks on the Swiss land border or for flights to and from other Schengen countries. There can be temporary or intermittent checks, however, so crossing the border without a passport is probably not a good idea. Documentary requirements for crossing internal Schengen borders are not regulated by EU law, but many individual states require foreigners to have a passport or similar ID document.
When the child is born, you will want to apply for a consular report of birth abroad. You should apply for a passport at the same time.
You will also want to approach the Swiss or French immigration authorities to find out whether the birth needs to be reported to them, or whether your child needs any documents to remain in the country as your dependent (for example, the French titre d'identité républicain.