"I can't eat any sweets" is a good idiomatic way of saying "I must not..." in American English. "I must not" sounds very stiff and fussy.
'For my health, I should not eat any sweets. ' Why isn't it acceptable?
Because there's emphasis from aucune, so it translates as must instead of should.
Should would be "devrais". It's the conditional tense which isn't used here.
The placement of the "ne" bothers me because it looks like the negation applies to "devoir" instead of to "manger". How would you say "I don't have to eat any sweets" ? (Where is Monsieur Sitesurf at Level 6?)
Why isn't cannot accepted instead of must not?
For my health, I cannot eat any sweets.
You physically can, but it's not a good idea. So 'can' is wrong.
for my health i don't have to eat any sweets