I don't know of a general proof. I know Euler had a conjecture that two orthogonal Euler squares of order 10 could not be found. That conjecture was disproved when a 10x10 pair was found (but a pair is not enough to create lunch bunches). Here is the reference.

Update: A lunch bunch arrangement is a resolvable block design of the form (n^2, n, 1), which is also known as an affine plane. An affine plane of order n exists if and only if a projective plane of order n exists. In the projective plane article it states:

A finite projective plane exists when the order n is a power of a prime...It is conjectured that these are the only possible projective planes, but proving this remains one of the most important unsolved problems in combinatorics.

So if one could prove the above conjecture, it would prove that the only lunch bunches are of the order of powers of a prime.

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