Apple - Can volume booster software (like Boom) damage MacBook Pro's internal speakers?

Boom is most likely the culprit to your speaker failing. Boom uses algorithms to increase the overall volume of the track to a point without clipping. Pushing speakers beyond what they are supposed to do can and will cause damage to them. The reason the speakers have a maximum volume is that is the safest high fidelity sound they can produce.

Here is a review of how boom works algorithmically http://www.macobserver.com/tmo/review/boom_lets_you_turn_your_macs_volume_up_to_eleven

Note the overall wave form increase in the article. This will cause damage, clipping and distortion with prolonged use.

A bit of background as to how this type of software can blow your speakers.

Boom is a compressor, in fact a brick-wall limiter.
See https://en.wikipedia.org/wiki/Dynamic_range_compression

What these devices do - & they are used a lot in the music industry to even out sound levels; sometimes tastefully, sometimes to try to gain another step in the Loudness War <- see my own answer on Music Fans for a not-too-technical explanation)
They do this, not by making the loud bits quieter, but by making the quiet bits louder - this increases the apparent loudness of the overall signal.

Music using extreme compression does sound louder, at the expense of any dynamic range - loud & quiet.

…and here is where that is no good for your speakers…
A speaker is a magnet with coil of wire wrapped round it, attached to which is a cone of paper or plastic [the speaker itself]
If you run an alternating current through that, it makes the coil move & the speaker sound.
The energy supplied to the cone is dissipated as the cone 'wastes' that energy by pushing & pulling at the air in front of it - that's how you can hear the sounds, of course.

But... not all the energy is transmitted as sound, there is wastage & that wastage is heat.

Your speakers were designed to be able to dissipate the heat of a normal music stream - Radio, CD, MP3, etc - even on full volume. Add software to make that speaker move far more than it was ever designed to do & the heat build-up will eventually melt the wire in the coil, if it doesn't actually vibrate the entire structure to breakage first.
It takes much more energy to produce bass frequencies than treble frequencies, so in a 2-way speaker system, the bass is just more likely to give out first.

my 2010 macbook pros's subwoofer speaker broken after 1 years I installed iboom, the thin membrane just ripped away. I would not suggest people installing iBoom