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It looks very nice!

That looks right!
Does anyone here make a square knot when stringing?

Here it is:
74382

I think it's a double overhand knot. I use this knot when there is a larger than normal drill hole that I'm afraid the regular overhand knot will get lost inside.
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I suspect you were thinking of larkshead knots, but I don't use those either. They look too asymmetrical to me.

Actually, no-- I don't use half hitch knots, ever, when knotting pearls. Just regular overhand knots (and sometimes doubled a..k.a jeweler's knots.) When finishing an endless strand I make a...

If the knots aren't holding, and it's for a client, I'd go ahead and use a little glue. Better safe than sorry.