No, in fact it doesn't matter who opened the box first.
Imagine we both open our boxes at around the same time, and communicate each other results at the speed of light.
- From my point of view, I opened the box first, got red and caused you to get blue. Your message confirms my hypothesis.
- From your point of view, you opened the box first, got blue and caused me to get red. My message confirms your hypothesis.
We can repeat this experiment thousands of times, every time, we both get a 50/50 chance of red vs blue, and every time, the hypothesis is confirmed.
In order to know who really was first, the only way is to wait for the other person result. For example: when you open the box, start a timer, stop it when you receive my message. It the time is less than the times it takes for light to travel between us, I was first. But in order to have this information, you have to wait, so it is not faster-than-light anymore.
Imagine we both open our boxes at around the same time, and communicate each other results at the speed of light.
- From my point of view, I opened the box first, got red and caused you to get blue. Your message confirms my hypothesis.
- From your point of view, you opened the box first, got blue and caused me to get red. My message confirms your hypothesis.
We can repeat this experiment thousands of times, every time, we both get a 50/50 chance of red vs blue, and every time, the hypothesis is confirmed.
In order to know who really was first, the only way is to wait for the other person result. For example: when you open the box, start a timer, stop it when you receive my message. It the time is less than the times it takes for light to travel between us, I was first. But in order to have this information, you have to wait, so it is not faster-than-light anymore.