Processing OBPSK as OQPSK
OBPSK in the way that you describe it is also called $\pi/2$-BPSK meaning that the signal constellation is different from bit to bit, rotating by $\pi/2$ after it is used. There are two variations of this format.


Use $\pm \cos(\omega_0 t)$ to transmit one bit and 
$\mp \sin(\omega_0 t)$ to transmit the next bit. Repeat this over and over again.  Note the $\mp$ before the $\sin$ carrier.
This method rotates the constellation by $\pi/2$ counterclockwise in going from $\pm \cos(\omega_0 t)$ to $\mp \sin(\omega_0 t)$, and clockwise when going from $\mp \sin(\omega_0 t)$ to $\pm \cos(\omega_0 t)$.
Use

$\pm \cos(\omega_0 t)$ to transmit the first bit
$\mp \sin(\omega_0 t)$ to transmit the second bit
$\mp \cos(\omega_0 t)$ to transmit the third bit
$\pm \sin(\omega_0 t)$ to transmit the fourth bit,
and repeat this over and over. Note the $\mp$ and $\pm$ signs. This
corresponds to the constellation rotating by $\pi/2$ counterclockwise after each and every bit transmission.





Neither form of OBPSK signal can be demodulated very well
using a OQPSK demodulator, though it would be possible to do
so. In OQPSK,
even-numbered bits are transmitted using $\pm\cos(\omega_0t)$ and
odd-numbered bits using $\mp\sin(\omega_0 t)$ but the important
difference is that the $2n$-th bit is transmitted during the
time interval
$[2nT, (2n+2)T)$ while the $(2n+1)$-th bit is transmitted during
the time interval $[(2n+1)T, (2n+3)T)$, i.e. in overlapping but offset time intervals.
At the demodulator, the matched filters/correlators in the I branch
and Q branch thus integrate/correlate over offset time intervals,
and the demodulated data bits come out of the OQPSK demodulator alternately on the I and Q lines, one every T seconds. 
On the other hand, in
OBPSK, the $2n$-th bit is transmitted during $[2nT, (2n+1)T)$ and
the $(2n+1)$-th bit during $[(2n+1)T, (2n+2)T)$. If demodulated with
a plain vanilla OQPSK demodulator, during $[(2n+1)T, (2n+2)T)$ the matched filter/correlator input in the I branch would be just noise 
(and similarly for the Q branch).
This will decrease the effective SNR. So, Yes, a standard OQPSK
receiver could be used to demodulate an OBPSK signal, but
there are better ways of demodulating an OBPSK signal than just
processing it through an OQPSK receiver.