Don't open it for Output: that specifically deletes the file if it exists and opens a new one: use OpenMode.Append instead.
But ... I would suggest that rather than keeping the data in a flat file, use a "defined format" file such as CSV, XML, or JSON - the latter is trivial to support a list of customers and can serialize and deserialize the entire list of instances in a single line of code if you use NewtonSoft.JSON: you can add it to your project via the NuGet Package Manager (Tools ... NuGet Package Manager ... Package Manager Console):
PM> Install-Package Newtonsoft.Json
And then just one line of code does all the work:
MyCustomers = JsonConvert.DeserializeObject<List<Customer>>(File.ReadAllText(path));
Plus, you get a file format that is usable directly in many other applications.