I'd go with yes.
Generation ships are by their nature designed to be remain within the same operational parameters for the entire mission with some safety margin added. So effective generation system should be as static as possible for the designer to achieve.
There are workarounds. I recently asked about a bubble-ring model of generation ship that would get around this by having enough redundancy that some social experimentation at a ring level would be practical. But that requires on entirely different model for how you are using the generation ships so it doesn't apply to your question.
An optimally static society would have no real need for social mobility. People would be tested for aptitude and then trained for tasks that suit them best.
Since the crew would be chosen for high ability at their tasks and high ability needs to be available for the following generations as well, it is reasonable to assume the original crew was chosen for high natural aptitude at their specific tasks. The eggs would presumably also be as much as possible be chosen for natural aptitude since that is all that can be done on Earth.
Since the crew was chosen for aptitude at specialized tasks and children socially take after their parents and the ship would be too static, unless something goes catastrophically wrong, to need social mobility it is reasonable to think the crew positions would be nearly hereditary. Specifically there would be division between leadership, management and technical positions. Something resembling the officers, NCOs, and the crew division of military systems. If it works and all that.
Once landfall is made the need for static society would not really change. You'd simply have vastly increased need for management and technical positions. Thus leadership positions would remain in the hands of the ship officers and the eggs would have been chosen for aptitude at the management and technical positions and be used to rapidly increase those classes of people. This would create a need to rapidly educate and train large numbers of people which would continue the ship tradition of training people to specific positions based on aptitude.
So the castes would probably be largely hereditary initially with some limited mobility to account for the fact that even with genetics and upbringing people are probably not that predetermined. But the division would not be between the crew and colonists rather it would be a continuation of the rigid social system on the ship into which the new born would be raised into.
Two notes though:
First, such a caste system would actually be fairly good for the needs of the initial colonization. Democracy and freedom are required for efficiently deciding what to do. When you have an abundance of things to do with limited resources and probably a fairly specific plans on how to do them, a society with strict hierarchy works just fine. So the caste system might actually be an advantage, not something to avoid.
Second, as the demands of the society change, the social structure will automatically evolve to match the needs. There will be friction, but generally you can assume that once the caste system is not useful, ie the amount of resources grows so there is some need to think about the direction of the colony not just survival, the society will move on. And unlike aristocrats of our own world, the crew would have access to information telling them about the inevitability of social evolution. They'd probably even have plans drawn by some very smart people back on Earth about how to handle the transitions to more liberal society.