How do I verify a Name Server?

I signed up for a web hosting service who gave me my own name servers.
How can one check to make sure the name servers are visible to the
public and configured properly?

I have tried an online NSLOOKUP tool and when I specify one of the
name servers, the domain I enter returns expected results. When I try
the other name server, it says the name server is not found.

Does anyone know if an online tools that will check to make sure a
name server can be found? I cannot check by pinging the server using
the name because they have stealthed the servers to block ping
requests.

I guess what I am looking for is a whois where I can enter the
nameserver.

Thanks,

John