Some help understanding solenoids/flyback diodes

Hello,
I've got an antique phone that I want to wire up as a doorbell via Arduino. I've got a boost converter to kick supply up to 40V or so, then an H-bridge to flip the voltage as controlled by Arduino input. The phone has two solenoids connected in series, with about 30ohm resistance, haven't measured the inductance yet.

I copied this circuit from the sparkfun product here:

My questions:

  1. In this configuration, I can't use a flyback diode because the voltage flips direction, right?
  2. If I'm using 5V supply instead of 3.8V, do I need to adjust the values of R3/C4? I'm having some trouble understanding the data sheet.
  3. Any other comments?

Thanks for the help, I have an understanding of how inductance works but haven't worked with boost converters/solenoids before.

That circuit uses an H-bridge chip (NJM2670) with free-wheel diodes integrated into it, it doesn't mind inductive load.

The ringer inductance will be substantial due to its size, able to store enough magnetic energy to trash
any unprotected driver circuit - the freewheel diodes allow the current to subside safely on switch-off, preventing
the 100's to 1000's of volts spike youd get otherwise. Note that large inductors can deliver fatal shocks, careful
playing with them.

Thank you for that, as I look at the block diagram I do see the diodes. Thanks also for the warning about the high voltage-- I'll go slow and careful.