Adding a clustered index on a key on which there is already a primary key constrain is an unnecessary duplication. Instead the primary key constrain should also be the clustered index. However, your question is actually different...
is the CustomerId column a suitable candidate for a clustered index?
One cannot answer this without knowing how you will be querying the table. There are numerous query patterns for which this organization will not be optimal (typical examples being time series where the time column is the appropriate clustered key). And there are just as many examples of queries for which this is the best clustered index. Unanswerable.
given that the ID field is the primary key, will SQL Server still need to 'uniqueify' the clustered index?
If the index is not declared unique then SQL Server will add the uniquifier column. The column value will never materialize though, as no duplicates will ever occur.
I have a 'Messages' table, which is used in a chat application ... WHERE CustomerId = @Id ... the ID field is the primary key
Pardon me, but this does not make any sense. What you're saying is that the Messages table can have only one message from each customer. That would make a horrible chat experience. I'm pretty sure your explanation is wrong.
I would expect a Customers table with clustered index and primary key constraint on CustomerId. The Messages table is likely to be organized by chat room, or some other group organization that pairs the chat participants. If the chats are always between one and exactly one Customer and a representative, then the 'chat room' may be the customer itself. In any way, the typical querying of such a Messages table would want all the Messages exchanged in a chat room, in the order posted or all the Messages exchanged with a customer, in the order posted. This is, actually, a partitioned time series and best served by a clustered index like (chat_id, post_time) or (customer_id, post_time). Note that this is not the primary key, the table may well have a message_id as primary key, but non-clustered.