Hi, Ron, gee, you sure know how to ask a question. I had to ask my dad (he's a farm boy as well as a biologist so that was handy) and he had to go away and look it up and we almost rang the National Museum!
Yes, we have what WE call horse flies but they are not what you call horse flies (from what I can make out). We have botflies, which attack horses (they don't lay eggs in wounds, they lay them on the horse's skin and then the horse licks the skin and they go inside the mouth or intestine and the grubs come out when they're hatched). We have two species of these that attack horses and one species that attacks sheep and goats.
They're all introduced cause originally there were no horses or sheep or goats here for them to live on!
We don't have your horse flies (tabanids). Thank goodness, they sound totally ick.
Our greatest problem is fly strike on sheep but that's caused by the common house fly (also introduced).
Sorry about the long post but having found all that out I was determined to get it all down, tee hee. I am now very well educated on flies. I'm so glad there are no kittycat flies.