I think your examiner's point in specifying a key is to get you thinking about writing in the friendliest way possible for the musician consuming your music. To that end, yes, raising notes on the way up & lowering them on the way down is a good rule of thumb, but one must also temper that with the need to introduce as few accidentals as necessary.
Having a key sets some initial parameters, namely 7 fixed notes, so ideally we need only inflect 5 of them to achieve a chromatic scale. Keeping to these rules — raise notes when ascending, lower when descending, inflect notes only when necessary — yields this chromatic scale on E♭:
E♭-E♮-F-F♯-G-A♭-A♮-B♭-B♮-C-C♯-D-E♭ | E♭-D-D♭-C-C♭-B♭-B♭♭-A♭-G-G♭-F-F♭-E♭
...which yields 5 accidentals in each direction; although, B♭-B♭♭-A♭ is more easily parsed when written B♭-A♮-A♭ — a change which adds one accidental to the descending scale. This is certainly easier to read with a key signature of B♭, E♭, A♭ than introducing notes like G♯, which adds extra accidentals to the scale & severs any relationship to E♭ Major.
As an aside, it may be an outmoded idea now, but i learned that there are 2 forms of the chromatic scale: melodic & harmonic.
The melodic form on E♭ in the key of E♭ Major is that described above.
The harmonic form is related to equivalence of the major & minor modes, and near-neighbour chords.
Combining E♭ Major & E♭ minor yields:
The missing lowered super-tonic & raised sub-dominant are provided by chords that resolve to the dominant:
V of V (
II) providing the raised sub-dominant (F-A♮-C) & the "Neapolitan" (
II♭) providing the lowered super-tonic (F♭-A♭-C♭).
Like the harmonic minor scale, the harmonic chromatic scale is the same in both directions, hence:
E♭-F♭-F♮-G♭-G♮-A♭-A♮-B♭-C♭-C♮-D♭-D♮-E♭ | E♭-D♮-D♭-C♮-C♭-B♭-A♮-A♭-G♮-G♭-F♮-F♭-E♭
Digression aside, i think the solution is to leverage your key signature & only use accidentals where necessary. So use the A♭ & don't introduce a note for which you already have an enharmonic equivalent in the key.