Just FYI...
Urchin spines are calcareous, and pee isn't likely to break those down --besides, yuck!
The pee thing IIRC is actually good against nematocysts from jellydish and other stinging coelenterates.
For urchin spines embedded in the skin, try a lemon juice or even vinegar soak. Don't try to worry the spine out, the spine can be actually quite filthy, and you don't want to grind it further into your flesh, haha.
Swelling can come as a reaction to this filth and/or venom delivered by the hollow spine.
With Diadema urchins, there's this maroon-black pigmentation on the spines that can remain in the puncture wound and APPEAR like a bit of spine. The lemon or vinear soak can reduce that stuff too.
HTH