Let me split this question into two questions. Can other anthrahydroquinone be used for production of hydrogen peroxide? Why is 2-ethylanthraquinone used for the commercial production of hydrogen peroxide?
The production of hydrogen peroxide can be done just as easily with a number of anthrahydroquinones including the parent anthrahydroquinone.
In order to produce a large amount of hydrogen peroxide, the anthrahydroquinone must be recycled. If not, it would be more expensive than hydrogen peroxide or rather, the cost of hydrogen peroxide would reflect the cost of the anthraquinone. Certainly, 2-ethylanthraquinone is probably lower melting and more solution in organic solvents than anthraquinone itself (just as toluene is lower melting than benzene). However, that is not the reason for use 2-alkylanthrahydroquinones. It is the recycle step that is critical. The production of hydrogen peroxide is a batch process. The electrocyclic reaction of oxygen with the anthrahydroquinone takes place first, the anthraquinone is isolated and reduced back to the hydroquinone. That is the critical step.
The reduction can produce a tetrahydroanthrahydroquinone
(some structures are wrong in this reference). While this can also still produce hydrogen peroxide, the quinone reacts with some hydrogen peroxide to produce an epoxide. That epoxide cannot be recycled. The hydrogenation reaction, when reducing the quinone, over-reduces the unsubstituted ring first. So, the substitution is helpful in preventing over-reduction in the recycle step.
I searched for "anthrahydroquinone peroxide" to find information. That search also discovered this reference:
Applications of hydrogen peroxide and derivatives
By Craig W. Jones
Published by Royal Society of Chemistry, 1999
ISBN 0854045368, 9780854045365
I was able view useful information in a Google books result of the search.