Can Naegleria fowleri enter through the eyes ( example rinsing/splashing eyes with water)
In a 1987 article "Nonenteric Infections Acquired through Contact with Water", the author mentioned that "Infection of the ears, throat, respiratory tract, and cornea are also encountered" by virulent strains of Acanthamoeba and Naegleria fowleri. The only reference that implicates Naegleria fowleri is this one, but it doesn't mention modes of infection.  

The long and the short of it is that Acanthamoeba infections can be acquired through contact with the cornea, but it's not clear whether Naegleria infections can also be acquired in this way.  

n.b. The first link takes you to the JSTOR page.  In the event you do not have access, here is the PubMed page.