[talanum1 (OP)]

For massless particles we have:

E^2 = (pc)^2.

This has solution:

E = +/- pc.

So why does nature choose only positive values?

This is at odds with:

E = |p|c.

[Bored chemist]

It doesn't.
What you get isn't an absolute energy, but a change in energy (associated with a change in mass).
And a change in energy is positive or negative depending on which way you look at it.

[talanum1 (OP)]

And a change in energy is positive or negative depending on which way you look at it.

That's not compatible with:

E = |p|c

even if p is change in momentum: "sizeof a vector" is always positive.