The Kinect is classified as a class 1 laser device (Review from Anandtech), which means the maximum emitted power of the laser is <25 μW. This amount of power, according to current knowledge, is safe. It should not be able to cause lasting damage, it doesn't necessarily mean it is not capable of causing afterimages or similar effects.
According to the OpenKinect page the IR laser diode itself is capable of emitting 60 mW at 830 nm, which is far above safe limits. The signal is spread out, which reduces the intensity to safe levels. There seem to be some safety features built-in to ensure that. If those safety features would fail it would be theoretically possible for the Kinect to emit a harmful amount of IR. I cannot say how likely that is, though.
There is also an image of the Kinect IR pattern in the review, but it does not look like the pattern you described. You can see that pattern yourself using e.g. a smartphone camera or any other camera without an IR filter.
We humans also are not capable of seeing electromagnetic emissions in the IR range. But emissions in the near IR range (800-1400 nm) are still focused onto the retina and can damage it. Additionally, an IR laser will not activate the blink reflex, which normally would prevent further damage to the eyes. There are also no pain receptors in on the retina that could warn you, at the point were you notice the laser it will be too late and permanent damage has been inflicted.
I would personally doubt that the Kinect is responsible, but I'm not really qualified to estimate the probability of the safety measures of the Kinect failing and to judge the medical consequences of unsafe amounts of IR emissions.
I am not a doctor, this is not medical advice and I strongly recommend you to have your eyes checked by a medical professional.
I own an Xbox 360 Slim bundled with Kinect for Xbox, and a Kinect for Windows. I tried plugging in the Kinect for Windows into the Xbox, and the console did NOT recognise it. So I can confirm that the Kinect for Windows will not work on the Xbox. On the other hand, the Kinext for Xbox -using the power adapter- does work on the PC.
Best Answer
No. You can not share a Kinect sensor for Xbox 360 with an Xbox One. As @Ramhound points out, the two iterations of the Kinect sensor do not even use the same connection.