In the commerce order checkout page, if you use "Save to my address book" checkbox in the billing address section, the address is getting saved to the user's address book. But if I am doing this using JSON request, how can I save the address to the user's address book and how can I load the saved addresses.
How you save depends somewhat on your desired UX. If you look at the code in the
BillingInformation checkout pane (in the checkout module) or the
CustomerProfile inline form plugin (in the order module), you'll see some properties used to instruct Commerce how and when to copy addresses to the addressbook.
If it's fine for your addresses to be copied when the order is placed (i.e. not immediately upon them being entered on a checkout form), then you can simply get by with setting
copy_to_address_book in the profile's data array to
TRUE. This will get picked up by the
AddressBookSubscriber (in the order module), resulting in the address being copied to the user's addressbook upon order placement.
Other options would be discoverable through the way the checkout pane functions. There is no way, as far as I know, to trigger an immediate copy merely by the save of the profile, but I haven't looked deeply at Commerce API to confirm.