There are two questions:
What makes a camera "professional", to a photographer?
What makes a camera "professional" to everyone else?
In this case these people clearly don't know anything about photography as their definition of "professional" is based on pretty stupid criteria.
For example: Detachable Zoom Lenses aren't allowed, so how about detachable prime lenses? Obviously the minimum wage highschool dropouts they'll have working security won't know the difference...
The definition of professional is nuanced. Anything a professional photographer uses, is professional gear, by definition. Professional means "someone who makes a living through photography." This has no implication about skill, or quality.
In terms of gear, professional "equipment" is gear designed for rugged use. It's built with very high quality control, to very exact specifications, highly durable, and offers a bevy of features that your regular hobbyist might not ever want or need. If you are someone who makes a living taking photos, and your cameras cannot ever be allowed to fail, then you buy a professional body. This is almost always reflected in the price, but if you need performance, you pay for it.
In the hands of a skilled photographer, any camera is capable of producing images that can grace magazine covers. fstoppers did a photo shoot using only an iPhone camera. And if some random person picks up a new 1Ds, there is a strong likelihood that the pictures will suck.
There will be many professionals who swear by non-professional grade bodies. And certainly they have a point. President Obama's Presidential Portrait was taken with a 5D Mark II. However the 5D II has only a 9-point AF system compared to the 1Ds Mark III 45 point AF system with like 36 high resolution cross-types at f4 or better.
So to recap: professional means professional grade: high durability, high standard of quality, crazy feature set, and full control.