Can I play Professor Juniper if it is the only card in my hand, and then draw 7 cards?
Yes, you basically discard your remaining hand of zero cards, then you draw 7 cards from your deck.
Source: Black & White FAQ (2011-04-28), PUI Rules Team (2011-04-28)

Relevant cards

Note: All applicable cards may not be listed below.
BW1_101