Sure you can find something. Car or van, it may be your choice (prices are probably different). Van - the 6-person MPV (e.g. Toyota) is common.
Fare, not sure. Here is my "educated" guess:
Regular taxi fare from Shenzhen Bao'An Airport to the border is about RMB $130, or HKD $150
Total is HKD $550.
With a private transfer, I would expect the price to be at least 3 times that. To may be 4 times.
Thus HKD $1650 to $2200 would be my guess.
Thanks for the info. Based on your guestimation I think I'll take the bus.