My first thought on reading your comment was to disagree and say no, we can have the exact value of 1, because we can choose our system of units and so we can make the square a unit square by fiat.

A better way to dispute the unit square diagonal argument for the existence of sqrt(2) would be to argue that squares themselves are unphysical, since all measurements are imprecise and so we can't be sure that any two physical lengths or angles are exactly the same.

But actually, this argument can also be applied to 1 and other discrete quantities. Sure, if I choose the length of some specific ruler as my unit length, then I can be sure that ruler has length 1. But if I look at any other object in the world, I can never say that other object has length exactly 1, due to the imprecision of measurements. Which makes this concept of "length exactly 1" rather limited in usefulness---in that sense, it would be fair to say the exact value of 1 doesn't exist.

Overall I think 1, and the other integers, and even rational numbers via the argument of AIPendant about egg cartons, are straightforwardly physically real as measurements of discrete quantities, but for measurements of continuous quantities I think the argument about the unit square diagonal works to show that rational numbers are no more and no less physically real than sqrt(2).