What parts of your system do you trust? You have to trust your clients 100%, because as soon as they have a connection string (which they should have), they can do anything in the database that they want. You must also trust the servers.
So it seems you trust everyone. This simplifies the security of the system: it is already safe, no matter how you distribute the connection string.
I have seen a lot of superstition when it comes to storing and distributing passwords and connection strings. Many people find it inconvenient to send and keep them clear. This is irrational because customers end up having it explicitly. It is impossible to prevent.
So, my advice: Make a simple web service providing the following API:
string GetConfigSetting(string name)
Clients can request this service for a connection string. This service is so simple that its interface is likely to never change.
In this case, it makes little sense to worry about encryption. The client application can be easily decompiled to access any decryption procedure. In addition, the client must ultimately decrypt the secret. At this point, the attacker controlling the client machine can read the secret explicitly.
source share