You can pair any Bluetooth earphone to your phone, including multiple Bluetooth headphones (but only use one set at a time).
I'm still thinking the Jaybirds are not going into pairing mode correctly. When most Bluetooth headphones enter pairing mode, they usually blink rapidly often in alternating colors to indicate they are actually in pairing mode.
Another thing to consider is to check with Jaybird about how many devices they can have been paired with. It's possible you have paired them the maximum number of devices they can be paired to and you may need to set them back to factory settings to get them to connect to your iPhone.