Unfortunately, the upgrade for monks in OA is not as clear as it could have been. This has lead to an apparent ambiguity in the rules.
The Players Handbook clearly states that flurry of blows can be used while armed with a monk special weapon. OA clearly states that the list of monk special weapons includes butterfly sword, club, jitte, kama, lajatang, nekode, nunchaku, quarterstaff, sai, three-section staff, and tonfa. Monks are not initially proficient with lajatang, nekode or three-section staff. The lajatang, quarterstaff and three-section staff are all large weapons.
The apparent ambiguity arises from the statement in the PHB indicating that the monk special weapons must be light. However, let us quote the sentence in full:
"The weapon must be light, so a Small Monk must use Tiny versions of these weapons in order to use the more favorable base attack."
Taking the sentence in context clearly shows that it is defining monk special weapons with regards to small sized monks only! If a monk is proficient with any of the weapons on the special weapon list he can use flurry of blows with that weapon. Common sense would show that even a western (i.e. not OA) monk should be able to use a quarterstaff with FoB.
As far as gaining the strength bonus for using a quarterstaff as a two handed weapon rather than a double weapon, no. Not with flurry of blows. Again, this is from the description of flurry of blows. When weapons are mentioned, it clearly says that if only one is held, the extra attack from FoB is either the weapon or an open hand attack, if two weapons are held, one is used for the normal attacks, the second is used for the extra attack.
However, a mnk using a quarterstaff at his UBAB can elect to use it in a two-handed mode and gain a bonus to his damage. It is only the FoB mode that requires it to be used in a double weapon mode.