Question
What are the conditions and criteria of the illness that prevents from Hajj?

Fatwa
All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His Slave and Messenger.

The illnesses and disabilities that prevent the obligation of Hajj on someone are: illnesses that hinder a man from travelling to Hajj or performing it.
If one has an illness that hinders him from obligatory Hajj, and it is hoped that he will be cured from it, then it is impermissible for him to authorize a proxy to perform Hajj on his behalf during his life.
If he is hindered by an illness that is terminal or a chronic, incurable disease, or if he is an old man who cannot travel without hard suffering; in all these cases he is considered unable to perform Hajj. If he has enough money and finds someone to perform Hajj on his behalf in compensation for fair, equivalent wage, then it is obligatory upon him to hire him to perform Hajj on his behalf on condition that the proxy has already performed Hajj for himself. This is the opinion of ‘Ali ibn Abi Taalib, may Allah be pleased with him, Al-Hasan, Ath-Thawri, Is-haaq, Ash-Shaafi‘i, Abu Haneefah, Ibn Al-Munthir and Daawood.
Maalik said: it is not obligatory upon someone to perform Hajj unless he is able to perform it in person. The preponderant opinion is the opinion of the majority of scholars. This is based on the Hadeeth of Ibn ‘Abbaas, may Allah be pleased with him, in which a woman from the tribe of Khath‘am said: "O Messenger of Allah! The obligation of Hajj enjoined by Allah on His slaves has become due on my father and he is old and weak, and he cannot sit firm on the mount; may I perform Hajj on his behalf?" The Prophet replied: “Yes.” That happened during the Farewell Hajj. [Al-Bukhari and Muslim]
If the crippled person does not have money to hire someone to perform Hajj on his behalf or has money but cannot find someone to hire or found someone who demands more than the fair, equivalent wage, then Hajj is not considered obligatory on him.
Allah Knows best.