If you are willing to switch to json.net, you can use the solution from Convert an int to bool with Json.Net.
If you wish to continue using JavaScriptSerializer, you will need to create a JavaScriptConverter for your MyObject type as follows:
class MyObjectConverter : JavaScriptConverter
{
    public override IEnumerable<Type> SupportedTypes
    {
        get { return new[] { typeof(MyObject) }; }
    }
    // Custom conversion code below
    const string myPropName = "MyProp";
    public override object Deserialize(IDictionary<string, object> dictionary, Type type, JavaScriptSerializer serializer)
    {
        object value;
        if (dictionary.TryGetValue(myPropName, out value))
        {
            dictionary[myPropName] = !value.IsNullOrDefault();
        }
        var myObj = new JavaScriptSerializer().ConvertToType<MyObject>(dictionary);
        return myObj;
    }
    public override IDictionary<string, object> Serialize(object obj, JavaScriptSerializer serializer)
    {
        var myObj = (MyObject)obj;
        // Generate a default serialization.  Is there an easier way to do this?
        var defaultSerializer = new JavaScriptSerializer();
        var dict = defaultSerializer.Deserialize<Dictionary<string, object>>(defaultSerializer.Serialize(obj));
        dict[myPropName] = myObj.MyProp ? 1 : 0;
        return dict;
    }
}
public static class ObjectExtensions
{
    public static bool IsNullOrDefault(this object value)
    {
        // Adapted from https://stackoverflow.com/questions/6553183/check-to-see-if-a-given-object-reference-or-value-type-is-equal-to-its-default
        if (value == null)
            return true;
        Type type = value.GetType();
        if (!type.IsValueType)
            return false; // can't be, as would be null
        if (Nullable.GetUnderlyingType(type) != null)
            return false; // ditto, Nullable<T>
        object defaultValue = Activator.CreateInstance(type); // must exist for structs
        return value.Equals(defaultValue);
    }
}
Then use it like:
        var serializer = new JavaScriptSerializer();
        serializer.RegisterConverters(new JavaScriptConverter[] { new MyObjectConverter() } );
        var json = serializer.Serialize(myObject);
Note - even though your MyObject class only has one property, I wrote the converter under the assumption that in real life it could have additional properties that should be serialized and deserialized automatically, for instance:
public class MyObject
{
    public bool MyProp { get; set; }
    public string SomeOtherProperty { get; set; }
}