Q:

REST - How would a PUT request handle auto-incremented resource identifiers

According to the HTTP 1.1. spec:

If the Request-URI does not point to an existing resource, and that
  URI is capable of being defined as a new resource by the requesting
  user agent, the origin server can create the resource with that URI.

So in other words, PUT can be used to create & update. More specifically, if I do a PUT request e.g.
PUT /users/1

and that user does not exist, I would expect the result of this request to create a user with this ID. However, how would this work if your backend is using an auto-increment key? Would it be a case of simply ignoring it if it's not feasible (e.g. auto-increment is at 6 and I request 10) & creating if it is possible (e.g. request 7)?
From the snippet I have extracted above it does appear to give you this flexibility, just looking for some clarification.

A:

I'd suggest that you use POST, not PUT, for an auto-increment key, or do not use the auto-increment key in the resource ID.
If you use POST, then you'd POST to /users rather than to /users/1.  The reply might redirect you to /users/1 or whatever the ID is. 
If you use PUT, then you might PUT to /users/10292829 where the number is a unique resource key generated on the client. This key can be time-generated, or it can be a hash of time, session ID, and some other factors to guarantee uniqueness of the value across your client audience. The server can then generate its own auto-incremented index, distinct from 10292829 or whatever.   
For more on that, see PUT vs POST in REST

Following up. . .
In the case of allowing PUT to /users/XXXXXXX, for all users, you'd end up with two distinct unique keys that refer to the same resource.   (10292829 and 1 might refer to the same user).  You'd need to decide how to allow the use of each of these different keys in a REST-style URL.  Because of the need to reconcile the use of these two distinct ids, I'd prefer to use the first option, POSTing to /users and getting a unique REST url of the created resource in the response. 
I just re-read the relevant section of RFC 2616, and saw a return code specifically designed for this in REST applications: 

10.2.2 201 Created
The request has been fulfilled and resulted in a new resource being created. The newly created resource can be referenced by the URI(s) returned in the entity of the response, with the most specific URI for the resource given by a Location header field. The response SHOULD include an entity containing a list of resource characteristics and location(s) from which the user or user agent can choose the one most appropriate. The entity format is specified by the media type given in the Content-Type header field. The origin server MUST create the resource before returning the 201 status code. If the action cannot be carried out immediately, the server SHOULD respond with 202 (Accepted) response instead.

So, the RESTful way to go is to POST to /users and return a 201 Created, with a Location: header specifying /users/1. 

