Answer:
No, the box jelly and the irukandji jelly are not the same. The box jelly is the most venomous because they can possibly kill a human in 3 minutes, have up to 15 tentacles, and they are named because they have a box-like shape to their bell.The irukandji on the other hand is the smallest jellyfish in the ocean at 2 cm, have four retractable tentacles and can cause death.
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Yes and no, the term box jellyfish applies to all cubozoans because of their box-like shape. Their bell is similar to scyphozoans, but somewhat squared off, with four short appendages (pedalium) projecting down from each corner. The two main types of box jellyfish are easily distinguished by how many tentacles are attached to each pedalium. Carybdeids have only one (or rarely, two or three) tentacles per corner, while chirodropids have many. So, your answer is yes, an Irukandji Jellyfish (Carukia barnesi), a thumbnail sized carybdeid with one tentacle per corner is a box jellyfish. Other carybdeidea species such as Malo kingi are also suspected of being responsible for the occasionally lethal Irukandji syndrome.
On the other hand, it is also common to use the term box jellyfish to refer specifically to the species, Chironex fleckeri, a basketball sized chirodropid with 15 tentacles per corner- sixty in all. This usage would mean that the answer to your question is no, they are not the same jellyfish. However, the confusion resulting from this use of the term, box jellyfish, in reference to only one particular species should be abandoned for the sake of clarity. The terms sea wasp and marine stinger are also commonly used for either one or all species of potentially lethal cubozoans, although not all box jellyfish are dangerous to humans.