IRF44n and push pull regulator topology
Your logic is flawed. For example, when Q1 is on, the drain of Q2 is forced to +24&nbsp;V by the autotransformer action of the primary winding. Similarly, when Q2 is on, the drain of Q1 is at +24&nbsp;V. The body diodes of the MOSFETs are never forward-biased.

One issue that you do need to worry about is leakage inductance on the primary winding, which stores energy that does not get coupled to the secondary. This stored energy can cause the drain terminals to rise higher than 2&times; the supply voltage, perhaps to levels that could damage the MOSFETs. It's usually a good idea to include a circuit that clamps the voltage at the ends of the primary winding to some level between 2&times; the supply voltage and the Vds(max) of the MOSFETs. This could be nothing more than a pair of zener diodes that break down at, say, 30-36&nbsp;V. Their power rating would depend on things like the actual value of the leakage inductance and the switching frequency.

On a project I once did, I was able to use a simpler solution. I was driving such a transformer with +175&nbsp;V, but I also had a +400&nbsp;V bus in the PFC circuit. I simply connected a pair of ordinary rectifier diodes between the transformer ends and the +400&nbsp;V bus, effectively "recycling" the energy which would otherwise have been wasted.

Note that with the PWM that your driver uses, there are also times when both transistors are switched off. Aside from the leakage inductance issue noted above, during such periods both ends of the transformer primary sit at +12&nbsp;V. This is a feedforward converter, not a flyback converter, which means that whenever current is flowing in the primary, there's also current flowing in the secondary, through the bridge rectifier. There's no significant energy stored in the transformer itself (i.e., it isn't "charged" and "discharged").