In your example of the Square Reader, it can be deceptive. I too wondered how there was power running to the headphone jack as Apple requires/uses the dock connector to charge, sync, etc. All the add-on devices use the dock and other than headphones I don't know of companies using the headphone jack. Why had no one used it before Square. Read on...
The Square reader actually takes the magnetic information from the swiped card and turns it into audio tones. These tones are then interpreted by the Square App and allow it to process the swipe like a regular credit/debit card transaction.
In my opinion, a very elegant programming solution.
This is the number one foreign object I've seen stuck in iPod / iOS headphone jacks and just like getting coins and business cards and CD's out of a stuck optical drive slot, your best bet is to get it in the hands of a trained technician to remove.
Now, if you can't get to a technician you trust, you'll have to use some ingenuity to decide if you are able to dislodge it.
If the bead rattles around when you gently invert the device, it's very likely you could get it out by mechanical means (centripetal force as you swing your arm) or light suction or very carefully getting a lever past the sphere.
Unfortulately, most cases the bead gets firmly wedged at the bottom due to force from a headphone jack and the force needed to crush the sphere usually damages the thin metal contacts that are designed to hold the tip of your headphone jack in the port.
You may want to consider the repair options and cost of a new jack before deciding to force out a foreign body such as a silica bead.
Good luck, and hopefully this doesn't end up in a costly repair or replacement for you.
Best Answer
I'd say one does not exist.
The reasoning behind you being able to find an adaptor from USB-C to lightning is that both are digital connections. I'm not sure what, if any, data translation would need to be done, but if it was necessary USB-C can carry the power to do it. (Similarly for the lightning to audio cable supplied with the iPhones).
To do the same from a regular headphone socket would need the adaptor to carry two-way DACs (digital analogues converters) to & from the earphones & mic, & would also have to be battery-powered, as it can't take power from the headphone socket on the Mac.
It's not impossible to do that, but it would be costly. It seems so far, no-one is willing to see if the consumer would be prepared to pay the price.