I agree one should always turned an integrated amp on last, after the source input device is turned on. If one forgets to do that, he may hear a pop, depending on the design of the specific amp, that typically wouldn't cause the amp itself to blow. It may do some damage to the speakers, if done repeatedly.
In the OP's case, I would agree that those Dyns are not easy loads, as they have specified sensitivity of 87 dB and impedance of 4 ohms. It sucks that they did not say 87 dB at 2.83V or 1W but let's assume it is at 2.83V just to be on the safe side. So that's equivalent to speakers specified as 8 ohms nominal with sensitivity at 84 dB at 2.83V, or 81 dB 1W/1M. Definitely not an easy load, but we are talking about a 300W/600W rated class D amp here. Now let's heed TLSGuy's remarks about class D amps (Note: not all class D amps have the limitations he referred to) for the moment, so let's de-rate his amp to something like 200W 8 ohms, 150W 4 ohms, just to be super conservative.
OP said he did not listen loud at all, he did sit 15 ft away, so let's further de-rate his amp output by another full 13 dB, that is without allowing for room gain and the effects of two speakers. That means he will get 68 dB from his listening position with the amp outputting just 1 watt!!
That means if what he meant by
was 75 dB or less of spl, then the amp would have been outputting about 5 watts !!! Add 15 dB of dynamic peaks that should be rare except for classical music and movies, the amp would still be outputting only 160 watts.
Note that the assumptions I made above are unrealistically conservative. In all probability, I would say the amp did not blow twice because of the very nice Dynaudio Focus 340 speakers, unless they are defective. We all know the OP is enjoying his same speakers with the Hegel H360. The H360 is just another integrated amp that is not overly powerful, so how likely is it that the Focus 340s had, and still have some sort of hidden defect? I highly doubt that, and I would love to have a word of the Dynaudio reps about their claim that those little 250 W IEC long term rated speakers could blow a 300/600 W rated class D amp, unless the said amp was poorly design or had some sort of defect.
Not knowing anything about that amp, and know very well Dynaudio's reputation and competence in speakers design, I know where I would bet my money on.