I'd be interested in seeing a picture of this. There is a concentric IR laser beam sharing the same axis as the green, so if you avoid the green you also avoid the IR as well. Am curious about the 'hidden' IR artifacts/emissions from these lasers though.
Better lasers have IR filters to block these things, but cheaper units don't unless you install it yourself.
My thoughts are that the IR square probably isn't overly dangerous due to its high divergence; figure a 300mW pump diode but the energy density in the square should be low compared to the laser beam. It's probably some sort of artifact similar to the "Osram square" that you see in the PL450 and PL520 diodes. I wouldn't put my face in it regardless, since it would be uncomfortable to look at if it were visible and it's hard to be certain as to just how much power is there. I wonder if the square is 808nm or 1064. I suspect 808.
One thing does need to be said about these lasers' IR emissions. I generally say that the IR is not dangerous due to the fact that most of the IR is concentric with the green beam, and as long as you avoid the green you avoid the IR as well. The green output is dangerous enough in itself that you will definitely want to avoid it regardless of whether your laser is IR filtered.
The IR emission only becomes a real danger, ironically, when safety glasses get involved. The green laser is generally stopped by short-wavelength safety glasses, i.e. the ones that are yellow or red in color. Unfortunately, such safety glasses usually don't block IR at all. As far as the IR is concerned, those safety glasses aren't even there. Unless the safety glasses are specifically rated to block both IR and short-λ, they will pass IR with no attenuation. That poses an eye hazard by definition, and it is a serious hazard once you get into tens of mW. Your blink reflex won't protect you either so that is a further problem.