Your question has been answered by our members. While we are sorry you are anxious, this scenario is not realistic.
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Quest diagnostics is a big operation. They, I am certain, follow protocol. It's a lab. They likely collect specimens, label them properly and then place them in the holding trays. Then those are collected when filled to be taken for the appropriate tests. In no way would this put any patients at risk.
What were you being tested for at the lab?
Quest doesn't give hiv to their patients going for blood tests. It is unrealistic for you to think that someone there doesn't know how to do their job and is reliant on you (who doesn't have any medical training) to tell them how to do their job.
Move on from this somewhat odd idea that you have discovered how medical staff should operate.