Does UDP Port 1434 need to be open for named instances with static TCP?
UDP 1434 is used for Named SQL Server instances and SQL Browser service listens on this port for any incoming requests to a named sql server instance. The browser service will respond to the client with TCP port no. for the requested named instance.

From BOL :


  The SQL Server Browser service lets users connect to instances of the Database Engine that are not listening on port 1433, without knowing the port number. 
  
  To use SQL Server Browser, you must open UDP port 1434. To promote the most secure environment, leave the SQL Server Browser service stopped, and configure clients to connect using the port number.


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  It needs to be open if the port is not explicit in the connection string?


Yes. If browser service is not running, you HAVE to specify servername,portNo e.g. YourSecureSQLServerName,50101