That's indeed correct. The Civil Rights Act of 1964 provides that service cannot be refused* on the "ground of race, color, religion, sex, or national origin", other than that it is really fair game.
In fact it is a right for businesses to choose who they want to serve. An airline that only flies costumed clients? 100% a right in this country. They can even decide who they consider costumed and who not.
* In research I discovered that it depends actually on the type of service. In California the Unruh Civil Rights Act catches all businesses; but federally only certain types of businesses must follow the discrimination rules. http://users.wfu.edu/zulick/341/civilrightsact1964.html
Airports are included in the Civil Rights Act because "it serves or offers to serve interstate travelers".
Note that in California the Unruh Act more or less states that businesses do not have a right to refuse service for anything not related to the business transaction. For example, the ACLU was successfully sued under the law for kicking off-duty police officers out of a seminar on police surveillance, and banks have been sued for denying loans to homeowners who expressed interests in running home-based day care centers.
You'll note that the government is inserted here, that Jetblue's determination that he couldn't fly was the result of government action. The government has a lot more rules about what it can't do to people than private business does.
As was noted in the article, JetBlue made that determination on their own. You can chose to believe the government told him no, but you'd be ignoring the stated facts in and just making shit up.
No, according to the second article they made their decision based on his state of mind after the government singled him out and messed with him. The government is directly responsible for that chain of events.
Any airline, any airpot, any business period can refuse to provide you service for any non-protected reason (race, gender, etc.).