You can only claim from either VHI or Quinn, not both.
You'll have to send original receipts. If you ask for receipts to be posted back to you after claim has been assessed, they will be stamped with something like "Quinn processed", hence can not claim again.
Once claim is settled with BUPA/Quinn, you'll be able to claim to revenue (form MED1) the difference between what you paid and refund from private health insurer at standard rate.
Both health insurers have a clause to deal with other insurance policies.
VHI - Rule 9f "If you or another member are entitled to claim under any other insurance policy for any of the costs, charges or fees for which you are insured under this contract, we will pay only our rateable proportion of these costs. When making a claim you must tell us if you have other insurance."
Basically, the point of health insurance is to cover the costs your incur for medical treatment. If you have two or more policies, then they will cover the costs, but you won't get anything extra.
When you were getting the treatment carried out, what health insurance details did you give to the provider/doctor/hospital? Or was it an out-patient service, such as a scan, lab test?
Finally, if there is part of the cost that you are paying yourself and that isn't covered by health insurance, you will be able to make a claim from Revenue.