Does a jailbreak void AT&T Next's upgrade?

According to AT&T's Next upgrade policy, my phone must be in "like new" condition in 18 months in order to get a new phone. I was told at AT&T that this meant that my phone couldn't be physically damaged in any way but the representative didn't mention anything about a jailbreak voiding the "like new" status.

Would jailbreaking my phone (iPhone 5s) void said "like new" status?

If you hand them the phone with it jailbroken then yes, it will void the policy. Just restore it before bringing it in and your all set.