Can dental hygienists have long nails?

I doubt it. Long nails can puncture gloves and make it harder to handle instruments and maneuver them in a person's mouth.

What? Really? What would you think if you were in the dentist's chair and the hygienist was fumbling around with sharp instruments in your mouth and couldn't get a good grasp on anything because her nails were too long? "Oops, sorry, I didn't mean to drop that probe into your throat. That must have hurt."

Mouths are a pretty small space for four hands to get into and do exacting work. Long nails are going to make it even more crowded.

No, they could interfere with the tools required in the job.