Yes, no and maybe.
You wouldn't be "disqualified" from health insurance if you had a pre-x, that is illegal. You could however not have that pre-x covered for a predetermined period of time if you have any sort of lapse in coverage.
All of this depends on whether your new coverage even has a pre-existing clause in it. The majority of employers don't elect the pre-x clause, at least in western NY. If that is the case, any condition would be fully covered regardless of lapse. If the policy does have a pre-x clause and you have a pre-x condition, you need to show you have had 18 months of continuous coverage for that condition to be covered.
Even if you were to lose insurance for a month or two, you would be offered COBRA by law as long as the employer has 20 or more employees. You would pay the full price of the premium (the employer would no longer subsidize it) plus up to a 2% administrative fee. Expensive, but a good option for people who need to prevent a pre-x.
I wouldn't say you have anything to worry about.