I did a quick google search and it seems there is a way to implement OnMouseEnter and OnMouseExit using the new InputSystem.
All you need to do is to have a PhysicsRaycaster on your camera and to implement IPointerEnterHandler and IPointerExitHandler interfaces in the HealthDisplay class.
The code than looks very similar, except it does not explicitly mention mouse anywhere:
public class HealthDisplay : MonoBehaviour, IPointerEnterHandler, IPointerExitHandler
{
[SerializeField] private Health _health;
[SerializeField] private GameObject _healthBarParent;
[SerializeField] private Image _healthBarImage;
(...)
public void OnPointerEnter(PointerEventData eventData)
{
_healthBarParent.SetActive(true);
}
public void OnPointerExit(PointerEventData eventData)
{
_healthBarParent.SetActive(false);
}
}