I didn't measure, but my guess is that the axle is 16mm in diameter. A 24mm wrench fits my nut perfectly. Parts fiche looks like they had a smaller sized nut across the flats, then upped it in '88. The axle size most likely stayed the same.
You are correct. The Honda CH250 (1985-1988) and the Helix (1986-87) used the 90305-ME5-003 ($2.97) axle nut which was 14.53 I.D. and 24.00 O.D. So your 24mm wrench will take the nut off and on. There was no Helix in 1988. The parts fiche shows 16mm which must be the axle diameter. The Helix (1992-2007, less skipped years) and the Reflex uses 90305-KBN-901 ($12.24 what the?) axle nut which has the same dimensions. What I learned in all of this is buy the $2.97 part because it's cheaper and will fit.
I just had mine off to change the tire and it was the largest in my set from Harbor Freight and I think it was 24mm. Mine is a 2006 and my memory of mm threads is not good as I was raised on sae.