Telling a child that they are an embarrassment is a huge psychological blow that will lead to internalized shame.

I think it's fine to joke about it ("OMG, you are slathering mayo on your fries. I can't be seen with you!"), but if we have true feelings of embarrassment, that's a time to look inward at our own shame triggers and figure out what's going on. Most likely, we feel frustrated or judged or worried about our parenting, and we're trying to put that on our children (who are living their own lives and finding their own way without regard for whatever visions we have in our heads).

One of the most terrifying but probably true things I've ever heard is, "Be careful how you speak to your children. That will become their inner voice."