You can pick up a bot scraper or knife for usually under $5, but a blunt serrated butter knife can do the trick too. You can also use a pair of hair scissors and remove them. It's better to remove them, yes, because horses can reingest them. Bot eggs are usually symptomatic of a need for deworming too.
I second the bot knife, but would also recommend deworming with Ivermectin after a couple of frosts have set. The problem with bot eggs is that they are ingested when the horse licks their legs and then the bots hatch into the stomach and this has been associated with gastric ulcers in horses. Ivermectin kills the hatched bots and doing it after a frost ensures there will be no more eggs for your horse to ingest.