Rules-as-written, the efreet cannot use wishes in this room, but this was not the rule or the intent of Tomb of Horrors releases for earlier editions of D&D.
AD&D
In the original AD&D Tomb of Horrors, the room does not explicitly prevent the efreet from using wishes. As in the 5th edition version, no spells or magic items work, except those that detect auras such as magic or evil, but this somehow does not prevent the illusion on 10,000 copper pieces in one of the chests, which have been enspelled to look like platinum.
In the AD&D edition, however, the efreet does not explicitly grant wishes, instead performing three "services", which may be implied to be wishes, as per the Monster Manual entry:
An efreeti can be forced to serve for a maximum of 1,001 days or by causing it to fulfil three wishes. They are not willing servants, and they will seek to pervert the intent of their masters by adhering to the letter of commands.
The word "wish" is not in italics, unlike the efreet's other spells, meaning that its ability to grant wishes in this edition was not considered a casting of the spell wish. Since the anti-magic only affects spells and items, the efreet in Gygax's Tomb of Horrors could originally grant wishes in the room.
D&D 3rd edition
In the earlier D&D 3rd edition version of Tomb of Horrors, the Efreeti has a unique special ability to allow its own special abilities, including wish. This explicitly resolves the conflict between the anti-magic effect, which in D&D is a variant antimagic field spell effect, which in that edition blocks spell-like abilities including the efreet's wish.
The illusion on the contents of the chests in the 3e Tomb of Horrors are also explicitly stated to be immune to the effects of the antimagic.
In short, the efreet in Bruce Cordell's third edition Tomb of Horrors could also grant wishes in the room.
D&D 5th edition
In this edition of Tomb of Horrors, no spells function within this room. The description of genie wishes in the Monster Manual specifies that genies specifically grant wishes by casting the wish spell. I strongly suspect that Jeremy Crawford, with his literal rules interpretations, would insist that this is the case.
The efreet is not bound to the room, and clever players might notice the antimagic and invite him outside after the first wish fails. However, the Monster Manual also notes that certain genies will pervert the intent of their wishes by adhering to the letter of the words, and the lawful evil efreet are most likely to do this, especially given earlier lore which says efreet are very specifically the ones who do this.
We have, in essence, a conflict between two statements in the same section: no spells work within the room, but the efreet does grant three wishes, and in earlier versions these were not considered mutually exclusive. Like all ambiguities in D&D, that's up to the DM to adjudicate, meaning of course that your approach was valid and within the rules. In fact, within the lore, the efreet may intentionally remain within the antimagic room just to cheat the player characters out of their wishes.
Best Answer
Here are what all the clues mean with room/area references.
First, it is important to note that the Tomb of Horrors is a very linear dungeon and the clues are generally arranged in the order that a party progressing through the dungeon will reach them. Some clues have been frequently misinterpreted by fans of the module, so I'll try to explain how the common misinterpretations that I have encountered are wrong.
Go back to the tormentor or through the arch, And the second great hall you’ll discover.
Shun green if you can...
but night’s good color Is for those of great valor.
If shades of red stand for blood...
...the wise Will not need sacrifice aught but a loop of Magical metal—you’re well along your march.
Two pits along the way will be found to lead To a fortuitous fall, so check the wall.
These keys and those are most important of all,
And beware trembling hands...
and what will maul...
If you find the false, you’ll find the true
And into the columned hall you’ll come,
And there the throne that is key and keyed.
The iron men of visage grim do more than Meets the viewer’s eye.
You’ve left and left and found my tomb
And now your soul will die.
OTHER RIDDLES:
In area 8...
In area 11...
In area 27A...