The dealership will recommend replacing parts that might not necessarily need to be replaced. Sounds like it may be a code 42 *alert alert*. Sounds like auto shutdown (ASD).
Are you thinking of doing it yourself? This I would recommend, along with getting a Haynes Manual. The neon's parts are very cheap. But if you do not have the time I would take it to a shop that is well respected, and get the work done there. This is just my experience with dealerships, and I've heard they make the majority of their money from maintenance(not sells). Plus neons are pretty easy to work on, especially compared to my last car, a 99 VW Passat.