It's not gender here, but number. "Either" and "neither" are singular. Added to which, "fruit" is often considered uncountable anyway (as in the saying "oranges are not the only fruit").
The example with "either fruit", though, doesn't really fit. "Either" implies there is a choice of one or the other, but you are actually saying you like both fruit (or "both types of fruit" -- "both" is plural). With the pizza, you should either say "either order" or "either of the orders". (The last one is short for "either one of the orders" -- it looks like it takes a plural, but it doesn't really.)
Also: "I like neither fruit" or, possibly, "I like neither of the fruit" (but the first sounds more natural to me).