I think I am going to give alot of criticism here but please don't take it too badly. I'm no expert but I have some things for you to consider.
First up class DBConnectClass.
The immediate thing which springs to mind is that you have hardcoded database connection information in the class. This immediately limits the class to one database. Of course you can make a copy of the file and edit the details, but then you've got two files to maintain instead of one. Why not pass the connection information to the constructor of the class for more portability.
Secondly you have a property $Connect which is only used in one method, and the class doesn't provide any kind of accessor for that property, so your relying on PHP's fallback mechanism. The property is unneccesary.
A very minor niggle is the naming convention. Why DBConnectClass - I already know it's a class. It might seem like nitpicking in the extreme but it's fairly important when you work in a team.
Next NavClass (naming again! :p )
What is the point of this class? It serves no purpose whatsoever other than wrapping require_once(). The files it works on are hard coded so again there is no portability. Really no need for this class to exist at all.
SuccessErrorClass (yeah you guessed it naming)
I think this is the real bone of contention which your prospective employer has. You have HTML intermingled with the class. Whilst this is sometimes unavoidable it is something to avoid as much as possible. As noted a key factor in good design is separation of business logic from presentation. The reason for this is when presentation and logic are intermingled it becomes difficult to make changes to the logic without having an impact on the presentation, and vice versa. Also when working with a team of people - commonly developers and interface designers - it makes sense that they don't clash with each other all the time. There is a technique or design pattern which deals with this problem called Model View Controller. Google knows all about it.
People have already suggested some good ideas on this so I won't go further.
My advice to anyone wanting to write object oriented code is to read as much as possible about inheritance, encapsulation and polymorphism. These are key to object oriented languages and important to have a good understanding of.
By the way inserting $_POST data directly into the database without any sanity checks is sure to make you look bad.
Good luck with the job 🙂