Is there anything in the Sunnah that indicates the prohibition of eating onions specifically on Thursday night and Friday during the day?

There is no specific prohibition mentioned in the Sunnah (definition) against eating garlic or onions throughout Friday night or day. Rather, the prohibition came against attending mosques for those who have eaten them, and this prohibition is absolute, covering every congregational prayer. The Prophet (peace be upon him) said: "Whoever eats onions, garlic, and leeks, let him not approach our mosque, for the angels are offended by what offends the children of Adam."

Umar ibn al-Khattab (may Allah be pleased with him) also said: "O people, you eat two plants which I consider nothing but vile: these onions and garlic. I saw the Messenger of Allah (peace be upon him) when he found their smell from a man in the mosque, he would order him to be taken out to al-Baqi'. So, whoever eats them, let him cook them thoroughly."

Eating raw onions is disliked due to their unpleasant odor when attending mosques. However, cooked onions are permissible to eat because their smell dissipates. And whoever eats garlic or onions and removes the smell by any means, there is no sin upon him in mingling with people in mosques and gatherings of goodness.