That looks like a very useful tool. Is there any advantage in generating a SegWit address over a regular one?
Yes. The only way we can see Segwit's at full capacity and lower fees is if more people start sending Segwit transactions. Current only ~9% of all Bitcoin transactions is using Segwit.
I am making a paper wallet for someone else to store funds so I want to make sure they can access them later in the future. I know for cold storage paper wallets I should just need an address and private key right?
Yes. You need the Address to fund it with coins and the private key to spend them. This wallet from Coinbin also lets you see your private keys, so you can use it on any other wallet software.