Do lower animals have instinctive behaviors?
Short answer
'Lower' organisms like C. elegans exhibit innate behaviors. In fact, unicellular organisms [without any nerve cells at all] can exhibit instinctive behaviors.

Background
Let's start by defining the central terms here, namely Behavior and Instinct:

Merriam-Webster has a general definition of Behavior: 


  [A]nything that an organism does involving action and response to stimulation


And according to the Cambridge dictionary, Instinct is:


  [T]he way people or animals naturally react or behave, without having to think or learn about it.


And Merriam-Webster defines Instinct more elaborately:


  
  Largely inheritable and unalterable tendency of an organism to make a complex and specific response to environmental stimuli without involving reason  
  Behavior that is mediated by reactions below the conscious level
  


Hence, C. elegans' actions are indeed instinctive behaviors. 

An interesting distinction here is whether behaviors in simple organisms should be indeed called behavior, or rather be classified as [lower-order] reflexes. 
According to Merriam-Webster, a Reflex is:


  [A]n automatic, often inborn response to a stimulus that typically involves a nerve impulse passing inward from a receptor to the spinal cord and then passing outward from the spinal cord to an effector (such as a muscle or gland) without reaching the level of consciousness and often without passing to the brain, such as the pain-withdrawal reflex, or the knee-jerk reflex.


This is a typical schoolbook approach-based definition of a reflex. Strictly spoken C. elegans' behaviors would not classify as reflexes according to this definition, because invertebrates do not have a spinal cord. 

Nonetheless, sea anemones show protective withdrawal responses by retracting their tentacles and oral disc. Some may even detach from the substrate in response to a variety of aversive mechanical, electrical, or chemical stimuli. And earthworms show rapid withdrawal responses mediated by giant nerve fibers when subjected to unfavorable stimuli, that are in fact classified as reflexes (Smith, 1991).

Indeed, Wikipedia defines a Reflex as 


  ...[A]n involuntary and
  nearly instantaneous movement in response to a stimulus. Scientific
  use of the term "reflex" refers to a behavior that is mediated via the
  reflex arc.


Thus the question then becomes what is a reflex arc? The Medical dictionary defines a Reflex arc as:


  Route followed by nerve impulses in production of a reflex act, from peripheral receptor organ through afferent nerve to central nervous system synapse and then through efferent nerve to effector organ.


Hence, in all, instinctive behavior is appropriate general terminology. When in simple organisms the behavior can be directly linked to a responseive action to a sensory stimulus, the term reflexive behavior may be a more appropriate term than instinctive behavior as far as I can see.

To push the definition of instinctive behavior even further, we can conclude from all this that single-celled organism exhibit it too, as they can show simple instinctive behaviors, such as phototaxis and chemotaxis, where they change their direction of movement in response to light and chemicals, respectively (source: Khan Academy). 

Reference
- Smith, ILAR J (1991); 33(1-2): 25-31