Hi there and welcome to the forums! Some of the larger critter keepers (medium and up) have been used as good homes for Bettas. On the forum it is suggest to have them in a minimum of a 2.5 gallon tank, this makes it easier to clean and it also gives them some more room to swim :)
The critter keeper is fine for now but upgrading him would be great too! But if you keep up with changing his water, for a small keeper that size I think either two 100% water changes or three 50% ones will do each week. But if you keep up with them you'll have a healthy Betta, but in my opinion no, I don't think they make good permanent homes.
Anywho, welcome to the forums and I hope you enjoy it here! Remember you can ask us anything, no matter how silly you think it is ^^