Your device handler will need to be re-worked for Hubitat. The button capability implementation is different (see here). Also any reference to physicalgraph such as in physicalgraph.device.HubAction needs to be changed to to habitat. See this thread for more information on porting apps and device drivers to Hubitat.
As for pairing, the Hubitat does not present Hub Event logs with a zbjoin event. When you enter "Discover Devices" mode in the Devices page of the Hubitat web-interface, if the hub recognizes a Zigbee device, you should see its Zigbee ID listed, with a message that it's initializing. Do you see that? If not, then the hub isn't seeing the device being ready to pair, and you should double-check the pairing procedure of the device.
If you do see the Zigbee ID listed, but the initialization never completes, there's a workaround I came up with to manually add a device. It involves copying down its Zigbee ID when displayed at pairing, and also keeping open the Zigbee Logging page that @ogiewon mentioned. In the logging page, a device being paired will first show up with its 4-digit hexadecimal DNI (Device Network ID). Those two pieces of information can be used to add the device manually using "Add Virtual Device".
For a complete list of instructions of this manual method, please see this post. I wrote it with Xiaomi Zigbee devices in mind, but it should work with any Zigbee device.
