Because then you have no way of identifying right angles. That leaves you with a claim you can make about right angles, and no angles you can ever apply the claim to. That is the same as not having the claim at all.
Consider this modified classic syllogism:
1. All men are mortal.
2. No one can ever know whether any particular object is or isn't a man.
3. Therefore, Socrates is mortal.
See the problem? It may still be possible to prove that Socrates is mortal, but you can't prove that or anything else with the postulate "all men are mortal", because that postulate, by hypothesis, doesn't apply to anything.
Of course, a theory that only includes the "equality axiom" would be incomplete, there would have to be other axioms that reference the (otherwise undefined) notion of the 'right angle'. Then it will be possible to prove theorems that involve this notion. For example, it will be possible to prove that "the particular object" that lies between the shorter sides of a triangle with side lengths three, four, and five, is, in fact, a right angle!
Until you have a definition of right angles, it is not possible to prove that anything, including the angle opposite the 5 side of a 3,4,5 triangle, is a right angle. Really, that's what a definition is.
It isn't necessary for the statement "all right angles are equal" to include a definition of right angles. But for the statement to have any meaning at all, the logical system that the statement is made in must include a definition of right angles. They cannot be left undefined.
Consider this modified classic syllogism:
See the problem? It may still be possible to prove that Socrates is mortal, but you can't prove that or anything else with the postulate "all men are mortal", because that postulate, by hypothesis, doesn't apply to anything.