If you look at an H-bridge, it resembles, to some extent, a differential amplifier.

A two-NPN-transistor diff-amp will sink current into its connected emitter(s) from whichever transistor whose base is more positive. If you bias the base of one transistor to somewhat above the "low" level, say by a couple of forward biased diodes, and put a pullup on each collector, then the collector will pull down the pullup on whichever transistor has the higher base voltage.

At the same time, if, rather than a resistor on each collector, there's a LED cathode and the other LED's anode. Rather than a pullup on each anode, there's a PNP's emitter. The two upper PNP's have their collectors tied together, at a pullup chosen such that the current when either transistor is on, will be suitable for the LED. If the two LED's need different currents, then the collectors must each have a properly chosen resistor.

Now there's an NPN at each cathode, and an NPN at each anode.

If the port pin drives an NPN at one LED's cathode, and also drives the NPN at its anode, then that LED will light. When that same port pin goes low, the opposite dif-amp base being biased to two or more forward-biased-diodes' Vf, the other LED will light.

Now, this is simplified, but you get the idea. I'd recommend using a quad transistor array, e.g. MPQ3904 or the like, actually. Those should allow you to operate with only 5 volts Vcc.

RE