Answer: Acanthamoeba species cysts and trophozoites. The diagnosis can be made by the characteristic morphology in conjunction with the clinical history. As mentioned by Florida Fan, the trophozoites have thorny or spiny projections called acanthopodia (from the Greek akantha meaning thorn), whereas the trophozoites of other amebae have rounded ends on their pseudopodia/projections.
The source (cornea) is also characteristic for this organism. Richard Garcia-Kennedy mentioned that not changing the contact lens case regularly is a common risk factor. Other important risk factors are cleaning contact lenses (or the case) with tap water and swimming while wearing lenses. I think I've scared all of my residents and Clinical Microbiology fellows out of these bad habits!
Like Rune, we have switched over to PCR for detection of all free-living amebae from clinical specimens. However, we used to use tap water agar (overlain with a bacterium like E. coli as a food source), and we would also occasionally supply our "cornealogists" with culture plates that they could directly inoculate. However we mostly had our clinicians send us specimens in MEM (minimum essential medium) which also worked quite well in supporting growth and viability until we could inoculate the specimen onto culture. When positive, corneal specimens and contact lenses usually grew Acanthamoeba within 24 hours (and in one memorable case, within 8 hours), but the PCR provides an even faster result and has proven to be just as sensitive as culture. It also allows us to detect Balamuthia mandrillaris (which won't grow on non-nutrient agar) and Naegleria fowleri (where time is of the essence for detection).
Thank you all for the comments on this case, and thanks again to Richard for donating it!