"I hadn't started mine yet, then I ripped mine open and it was more cooked than his one was, but it was still raw."
Mr Kinney, 20, said it was lucky Mr French had only had a couple of bites.
Eating uncooked chicken can lead to potentially lethal food poisoning through salmonella bacteria.
Mr Kinney took a picture of the sandwiches and posted it to Twitter, saying he hoped "neither of us wake up dead from eating raw chicken".
The pair took the burgers back to the Albany branch, and staff were "really apologetic" and gave them a refund.
On Tuesday, Mr Kinney emailed Burger King head quarters about the incident and received a reply later that day which offered him vouchers and said they were looking into their "chicken cooking procedures".
"Which I thought was funny because they obviously don't have any ... but I don't really want any vouchers because I never want to eat at Burger King again."
Burger King spokeswoman Rachael Allison said they had investigated the incident and established that the staff member who cooked the burgers had cooked the chicken on the wrong setting which gave it a shorter cooking time.
Ms Allison said Burger King deeply regretted the incident and had "apologised unreservedly" to Mr Kinney.