I'm having a bit of an issue with the names of the buttons on the switchbar.

I'm connecting to multiple servers which are not on any public groups, and mIRC is deciding that one of them should be labeled DALNet on the switchbar. Is there any way to force a change for that server?

mIRC gets the name from the lines:

Welcome to the [color:red]DALnet IRC Network [/color]

and

... PREFIX=(ov)@+ NETWORK=[color:red]DALnet SILENCE=10 ... are available on this server[/color]

So it's servers' fault, not mIRC's
All you can do is to ask servers' admin to change it (if you're sure it's not one of DALnet's servers).

There is also an option Connect -> Options -> "Multi-Server" and then "show in status window switchbar buttons", where you can disable showing of Network Name.

I really dont know maybe you should try to look it up and read abite and then if you read what it says in the read up it may be possible to find out how to do it