Essentially, yes, but there is a detail that you don't have quite right. HSA eligibility (covered under a High Deductible Health Plan) makes you eligible to contribute new funds to your HSA. However, it has nothing to do with spending money from your HSA. Once you have money in your HSA, you can spend it on medical expenses, whether or not you remain eligible to contribute new money to the HSA.
The medical expense clock begins when you first establish your HSA. Any expenses incurred before that date are not allowed to be paid from your HSA. But anything after that date can be reimbursed from the HSA, and there is no time limit for being reimbursed.
So yes, you can start saving all of your medical bills (even going back to when you first established the HSA). If at some point in the future you become ineligible to contribute new money to your HSA (because you became covered under an ineligible health plan), you can still save your medical bills for future reimbursement. You can let the money grow tax free in your HSA, and at some point in the future you can reimburse yourself for all the past medical expenses, tax free.
Note: The IRS gets notified each year when you take a distribution from your HSA. So 10 years from now, if you decide to empty your HSA all at once, you may hear from the IRS. That is why it is critical that you keep all the records related to this. Make sure that every penny that you take out of your HSA is matched up with a medical bill that you could show the IRS if asked.