There are a number of ways to make a high side driver that's required for Q1 and Q3, but generally they are either an IC or a number of discretes.
Here's an example:
I found it at: **broken link removed**. They don't tell you how to get VGH, which should be 7-12 volts above your top power rail, or in your case it would be 19-24 volts. It needs to have a generous peak current capability (0.1uF or more) but only needs to provide a few mA continuous. If you change R7, R10, and R11 to be 33k, it should still be fast enough for your circuit since you aren't changing directions frequently or rapidly.
[edit]The gate voltage should be independently, you can't hook two gates together.[/]
I wish I could offer a simpler solution, but this is why for one-offs I use the slightly more expensive P-channel MOSFETs instead.
[edit]Here's a thread with some simpler solutions. Post #10 from audioguru looks good. Note that it will only drive the MOSFET as long as the charge remains on the capacitor. This is a factor of the leakage in the diode, transistors, and of course the capacitor.[\]
In your application, I would seriously consider small relays.