That one's easy for me, but I probably wouldn't be able to find the answer on the Internet. I wrote my PhD thesis on Paramecium, which is another of those unicellular eukaryotes. It's covered with a thick protein coat outside its cell membrane. The protein is called the 'immobilization antigen,' and I wrote a paper called, "The immobilization antigen of Paramecium Aurelia is a single giant polypeptide chain." This was a big debate around 1970, because the main lab said it was a small protein, and some little group in Australia said it was a giant protein. I discovered the protein broke down into little pieces under the conditions that the main lab was using to isolate it. The guy who led the Australian lab was happy with my result ;-]

So, yes, unicellular eukaryotes export proteins, to give themselves a protective coat and maybe for other purposes, such as for signaling to other cells. I looked on the Internet but didn't find anything about this.

Thanks for your fun question!