You could create a custom ContractResolver that inserts a synthetic "_date_properties_" property into the contract of every object that is serialized.  
To do this, first subclass DefaultContractResolver to allow contracts to be fluently customized after they have been created by application-added event handlers:
public class ConfigurableContractResolver : DefaultContractResolver
{
    readonly object contractCreatedPadlock = new object();
    event EventHandler<ContractCreatedEventArgs> contractCreated;
    int contractCount = 0;
    void OnContractCreated(JsonContract contract, Type objectType)
    {
        EventHandler<ContractCreatedEventArgs> created;
        lock (contractCreatedPadlock)
        {
            contractCount++;
            created = contractCreated;
        }
        if (created != null)
        {
            created(this, new ContractCreatedEventArgs(contract, objectType));
        }
    }
    public event EventHandler<ContractCreatedEventArgs> ContractCreated
    {
        add
        {
            lock (contractCreatedPadlock)
            {
                if (contractCount > 0)
                {
                    throw new InvalidOperationException("ContractCreated events cannot be added after the first contract is generated.");
                }
                contractCreated += value;
            }
        }
        remove
        {
            lock (contractCreatedPadlock)
            {
                if (contractCount > 0)
                {
                    throw new InvalidOperationException("ContractCreated events cannot be removed after the first contract is generated.");
                }
                contractCreated -= value;
            }
        }  
    }
    protected override JsonContract CreateContract(Type objectType)
    {
        var contract = base.CreateContract(objectType);
        OnContractCreated(contract, objectType);
        return contract;
    }
}
public class ContractCreatedEventArgs : EventArgs
{
    public JsonContract Contract { get; private set; }
    public Type ObjectType { get; private set; }
    public ContractCreatedEventArgs(JsonContract contract, Type objectType)
    {
        this.Contract = contract;
        this.ObjectType = objectType;
    }
}
public static class ConfigurableContractResolverExtensions
{
    public static ConfigurableContractResolver Configure(this ConfigurableContractResolver resolver, EventHandler<ContractCreatedEventArgs> handler)
    {
        if (resolver == null || handler == null)
            throw new ArgumentNullException();
        resolver.ContractCreated += handler;
        return resolver;
    }
}
Next, create an extension method to add the desired property to a JsonObjectContract:
public static class JsonContractExtensions
{
    const string DatePropertiesName = "_date_properties_";
    public static void AddDateProperties(this JsonContract contract)
    {
        var objectContract = contract as JsonObjectContract;
        if (objectContract == null)
            return;
        var properties = objectContract.Properties.Where(p => p.PropertyType == typeof(DateTime) || p.PropertyType == typeof(DateTime?)).ToList();
        if (properties.Count > 0)
        {
            var property = new JsonProperty
            {
                DeclaringType = contract.UnderlyingType,
                PropertyName = DatePropertiesName,
                UnderlyingName = DatePropertiesName,
                PropertyType = typeof(string[]),
                ValueProvider = new FixedValueProvider(properties.Select(p => p.PropertyName).ToArray()),
                AttributeProvider = new NoAttributeProvider(),
                Readable = true,
                Writable = false,
                // Ensure // Ensure PreserveReferencesHandling and TypeNameHandling do not apply to the synthetic property.
                ItemIsReference = false, 
                TypeNameHandling = TypeNameHandling.None,
            };
            objectContract.Properties.Insert(0, property);
        }
    }
    class FixedValueProvider : IValueProvider
    {
        readonly object properties;
        public FixedValueProvider(object value)
        {
            this.properties = value;
        }
        #region IValueProvider Members
        public object GetValue(object target)
        {
            return properties;
        }
        public void SetValue(object target, object value)
        {
            throw new NotImplementedException("SetValue not implemented for fixed properties; set JsonProperty.Writable = false.");
        }
        #endregion
    }
    class NoAttributeProvider : IAttributeProvider
    {
        #region IAttributeProvider Members
        public IList<Attribute> GetAttributes(Type attributeType, bool inherit) { return new Attribute[0]; }
        public IList<Attribute> GetAttributes(bool inherit) { return new Attribute[0]; }
        #endregion
    }
}
Finally, serialize your example type as follows:
var settings = new JsonSerializerSettings
{
    ContractResolver = new ConfigurableContractResolver
    {
        // Here I am using CamelCaseNamingStrategy as is shown in your JSON.
        // If you don't want camel case, leave NamingStrategy null.
        NamingStrategy = new CamelCaseNamingStrategy(),
    }.Configure((s, e) => { e.Contract.AddDateProperties(); }),
};
var json = JsonConvert.SerializeObject(example, Formatting.Indented, settings);
This solution only handles statically typed DateTime and DateTime? properties.   If you have object-valued properties that sometimes have DateTime values, or a Dictionary<string, DateTime>, or extension data containing DateTime values, you will need a more complex solution.
(As an alternative implementation, you could instead subclass DefaultContractResolver.CreateObjectContract and hardcode the required properties there using JsonContractExtensions.AddDateProperties(), however I thought it would be more interesting to create a general-purpose, fluently configurable contract resolver in case it becomes necessary to plug in different customizations later.)
You may want to cache the contract resolver for best performance.
Sample .Net fiddle.